Introduction to Real Sequences and Convergence

This document from the National Polytechnic School (Preparatory Cycle-ENP), Math Analysis 1, by F. Fellag, provides a cheatsheet on real sequences.

Real Sequences: Definitions & Properties

Definition of a Real Sequence

• A sequence of real numbers is a function $u$ defined on a set of the form $\left\{n \in \mathbb{Z}, n \geq m\right\}$, where $m$ is a non-negative integer.
• The sequence is denoted by $(u_n)_{n \in \mathbb{N}}$.
• $u(n)$ is denoted by $u_n$, which is called the general term of the sequence.

Examples of Sequence Definitions

• Explicit Definition: $u_n = \frac{1}{n}$.
  • First three terms (for $n \geq 1$): $u_1 = 1$, $u_2 = \frac{1}{2}$, $u_3 = \frac{1}{3}$.
• Recurrent Definition: $\left\{ \begin{array}{c} u_1 = 2 \\ u_{n+1} = 3u_n + 1 \end{array} \right.$
  • First terms: $u_2 = 3(2) + 1 = 7$, $u_3 = 3(7) + 1 = 22$.

Bounded Sequences

Let $(u_n)_{n \in \mathbb{N}}$ be a real sequence.

1. Bounded above: $\exists M \in \mathbb{R}, \forall n \in \mathbb{N}: u_n \leq M$.
   • Example: $u_n = -\frac{1}{n}$ is bounded above by $0$ since $u_n \leq 0$ for all $n$.
2. Bounded below: $\exists m \in \mathbb{R}, \forall n \in \mathbb{N}: u_n \geq m$.
   • Example: $u_n = n^2$ is bounded below by $0$ since $u_n \geq 0$ for all $n$.
3. Bounded: $\exists m, M \in \mathbb{R}, \forall n \in \mathbb{N}: m \leq u_n \leq M$.
   • Equivalent definition: $\exists M \in \mathbb{R}^+, \forall n \in \mathbb{N}: |u_n| \leq M$.
   • Example: $u_n = \frac{1+n}{n+2}$ is bounded since $0 \leq u_n \leq 1$ for all $n$.

Monotonic Sequences

A sequence $(u_n)$ is called monotonic if it is either increasing or decreasing.

• Increasing: $\forall n \in \mathbb{N}: u_n \leq u_{n+1}$.
• Strictly Increasing: \forall n \in \mathbb{N}: u_n < u_{n+1}.
  • Example: $u_n = n^2$ is strictly increasing because (n+1)^2 > n^2.
• Decreasing: $\forall n \in \mathbb{N}: u_n \geq u_{n+1}$.
• Strictly Decreasing: \forall n \in \mathbb{N}: u_n > u_{n+1}.
  • Example: $u_n = \frac{1}{n}$ is strictly decreasing because \frac{1}{n+1} < \frac{1}{n}.
• Constant: $\forall n \in \mathbb{N}: u_n = u_{n+1}$.

How to Study Variations (Monotonicity)

1. Difference Method: Study the sign of $u_{n+1} - u_n$.
   • If $u_{n+1} - u_n \geq 0$, increasing.
   • If $u_{n+1} - u_n \leq 0$, decreasing.
2. Ratio Method: If all terms $u_n$ are positive, compare $\frac{u_{n+1}}{u_n}$ to 1.
   • If $\frac{u_{n+1}}{u_n} \leq 1$, decreasing.
   • If $\frac{u_{n+1}}{u_n} \geq 1$, increasing.

Monotonicity of Recurrent Sequences with an Increasing Function

Let $u_1 \in D$ and $u_{n+1} = f(u_n)$ with $f$ increasing.

• If $u_2 - u_1 \geq 0$, then $(u_n)_{n \geq 1}$ is increasing.
• If $u_2 - u_1 \leq 0$, then $(u_n)_{n \geq 1}$ is decreasing.

Example: $u_1 = -2$, $u_{n+1} = 3u_n + 1$. Here $f(x) = 3x+1$ is increasing. $u_2 = -5$. Since $u_2 - u_1 = -7 \leq 0$, the sequence is decreasing.

Limits of Real Sequences

Definition of Convergence

A sequence $(u_n)_{n \in \mathbb{N}}$ converges to $l$ (denoted by $\lim u_n = l$) if:

\forall \varepsilon > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow |u_n - l| \leq \varepsilon

• This means that for any small $\varepsilon$, all terms $u_n$ from $n_0$ onwards are within the interval $[l - \varepsilon, l + \varepsilon]$.

Example: To show $\lim \frac{n+1}{n+2} = 1$, we need to find $n_0$ such that for any \varepsilon > 0, $| \frac{n+1}{n+2} - 1 | \leq \varepsilon$. This simplifies to $\frac{1}{n+2} \leq \varepsilon$, which gives $n \geq \frac{1}{\varepsilon} - 2$. So, we can choose $n_0 = \lceil \frac{1}{\varepsilon} \rceil - 2$ (or similar). Given the source context example, it was $u_{n} = \frac {n}{n + 1}$ and $\lim u_{n} = 1$, then $n \geq \frac {1}{\varepsilon} - 1$ and $n_0 = [\frac{1}{\varepsilon}]$.

Divergent Sequences

• A sequence $(u_n)_{n \in \mathbb{N}}$ is divergent if it is not convergent.
• Divergence occurs if:
  • $\lim u_n = +\infty$
  • $\lim u_n = -\infty$
  • $(u_n)$ has no finite or infinite limit (e.g., $u_n = (-1)^n$).

Definition of Infinite Limits

1. $\lim u_n = +\infty$: \forall A > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow u_n \geq A.
2. $\lim u_n = -\infty$: \forall A > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow u_n \leq -A.

Example: To show $\lim (2n+1) = +\infty$, we need to find $n_0$ such that for any A > 0, $2n+1 \geq A$. This gives $n \geq \frac{A-1}{2}$. So, we can set $n_0 = \lceil \frac{A-1}{2} \rceil$ or $n_0 = [A]+1$ as in the source.

Properties of Limits

1. Uniqueness of Limit: If a sequence converges, its limit is unique. (Proof by contradiction: assuming two different limits $l_1, l_2$ leads to $|l_1 - l_2| \leq \frac{2}{3}|l_1 - l_2|$, which implies $l_1 = l_2$).
2. Boundedness of Convergent Sequences: If a sequence $(u_n)_{n \in \mathbb{N}}$ converges, then it is bounded.
   • Proof sketch: For $\varepsilon=1$, terms $u_n$ for $n \geq n_0$ are in $[l-1, l+1]$. The finite number of initial terms $\{u_1, \dots, u_{n_0-1}\}$ are also bounded, so the entire sequence is bounded.

Arithmetic Operations with Limits (for $\lim u_n = l$ and $\lim v_n = l'$)

• Scalar Multiplication: $\lim \lambda u_n = \lambda l$ (for $\lambda \in \mathbb{R}$).
• Sum: $\lim (u_n + v_n) = l + l'$.
• Product: $\lim (u_n v_n) = l l'$.
• Reciprocal: If $u_n \neq 0$ and $l \neq 0$, then $\lim \frac{1}{u_n} = \frac{1}{l}$.
• Quotient: If $v_n \neq 0$ and $l' \neq 0$, then $\lim \left(\frac{u_n}{v_n}\right) = \frac{l}{l'}$.

Limits and Absolute Values

• If $\lim u_n = l$, then $\lim |u_n| = |l|$.
  • Proof uses triangle inequality: $||a| - |b|| \leq |a - b|$. Applying this to $u_n$ and $l$: $||u_n| - |l|| \leq |u_n - l|$. Since $|u_n - l|$ tends to 0, $||u_n| - |l||$ also tends to 0.
• A sequence $(u_n)$ is a null sequence if $\lim u_n = 0$.
  • $(u_n)$ is a null sequence if and only if $(|u_n|)$ is a null sequence. This means $\lim |u_n| = 0 \iff \lim u_n = 0$.

Monotone Convergence Theorem

1. If a sequence $(u_n)_{n \in \mathbb{N}}$ is increasing and bounded above, then it converges to its supremum.
2. If a sequence $(u_n)_{n \in \mathbb{N}}$ is decreasing and bounded below, then it converges to its infimum.

Corollary: Every monotone and bounded sequence is convergent.

Example: The sequence $u_n = \frac{1}{n+1} + \dots + \frac{1}{n+n}$ is increasing and bounded above by 1, so it converges.

Limits and Inequalities

Theorems on Limits and Weak Inequalities

1. If $(u_n)$ converges and $u_n \geq 0$ for sufficiently large $n$, then $\lim u_n \geq 0$.
2. If $(u_n)$ converges and $u_n \leq 0$ for sufficiently large $n$, then $\lim u_n \leq 0$.
3. Limits Preserve Weak Inequalities: If $a_n \to l$ and $b_n \to l'$ and $a_n \leq b_n$ for sufficiently large $n$, then $l \leq l'$.

Remark: Limits DO NOT preserve strict inequalities. E.g., \frac{1}{n^2} < \frac{1}{n} but $\lim \frac{1}{n^2} = \lim \frac{1}{n} = 0$. So $l \leq l'$ holds, not l < l'.

Theorems on Infinite Limits and Inequalities

1. If $a_n \to +\infty$ and $a_n \leq b_n$ for sufficiently large $n$, then $b_n \to +\infty$.
2. If $b_n \to -\infty$ and $a_n \leq b_n$ for sufficiently large $n$, then $a_n \to -\infty$.

Example: To determine $\lim \frac{e^n + \sqrt{n^2 + 1}}{\sqrt{n + 1}}$. We have $\frac{e^n + \sqrt{n^2 + 1}}{\sqrt{n + 1}} \geq \sqrt{\frac{n^2 + 1}{n + 1}}$. Since $\lim \sqrt{\frac{n^2 + 1}{n + 1}} = +\infty$, the original limit is also $+\infty$.

Squeeze Theorem (Sandwich Theorem)

Let $(u_n), (v_n), (w_n)$ be three sequences. If for all sufficiently large $n$, $u_n \leq w_n \leq v_n$ and $\lim u_n = \lim v_n = l$, then $(w_n)$ is convergent and $\lim w_n = l$.

Example 1: To determine $\lim \frac{n(\cos(n) + \sin(n))}{(n + 1)^2}$.

• We know $0 \leq |\cos(n) + \sin(n)| \leq 2$.
• So, $0 \leq \left|\frac{n(\cos(n) + \sin(n))}{(n + 1)^2}\right| \leq \frac{2n}{(n + 1)^2}$.
• Since $\lim \frac{2n}{(n + 1)^2} = \lim \frac{2n}{n^2 + 2n + 1} = \lim \frac{2/n}{1 + 2/n + 1/n^2} = 0$.
• By the Squeeze Theorem, $\lim \left|\frac{n(\cos(n) + \sin(n))}{(n + 1)^2}\right| = 0$, which implies $\lim \frac{n(\cos(n) + \sin(n))}{(n + 1)^2} = 0$.

Example 2: To determine $\lim \frac{E(\sqrt{n}) + 1}{n^2}$.

• Using the property x-1 < E(x) \leq x, we have \sqrt{n}-1 < E(\sqrt{n}) \leq \sqrt{n}.
• Adding 1: \sqrt{n} < E(\sqrt{n}) + 1 \leq \sqrt{n} + 1.
• Dividing by $n^2$: \frac{\sqrt{n}}{n^2} < \frac{E(\sqrt{n}) + 1}{n^2} \leq \frac{\sqrt{n} + 1}{n^2}.
• Since $\lim \frac{\sqrt{n}}{n^2} = \lim \frac{1}{n\sqrt{n}} = 0$ and $\lim \frac{\sqrt{n} + 1}{n^2} = \lim (\frac{1}{n\sqrt{n}} + \frac{1}{n^2}) = 0$.
• By the Squeeze Theorem, $\lim \frac{E(\sqrt{n}) + 1}{n^2} = 0$.

Subsequences

• A subsequence of $(x_n)$ is formed by taking selected terms from the original sequence in their original order.
  • If $(x_n)_{n \in \mathbb{N}}$ is a sequence, and $\varphi: \mathbb{N} \to \mathbb{N}$ is a strictly increasing function, then $(x_{\varphi(n)})_{n \in \mathbb{N}}$ is a subsequence.
  • Example: $(x_{2n})$, $(x_{2n+1})$, $(x_{n^2})$ are common subsequences.
• Property: If $\lim x_n = l$, then any subsequence $(x_{\varphi(n)})$ also converges to $l$.
  • This is because $\varphi(n) \geq n$, so if $n \geq N$, then $\varphi(n) \geq N$.

Divergence Criteria using Subsequences

A sequence $(u_n)$ is divergent if:

1. It has two convergent subsequences with different limits.
   • Example: $u_n = (-1)^n$. $u_{2n} \to 1$ and $u_{2n+1} \to -1$. Since $1 \neq -1$, $(u_n)$ is divergent.
2. It is unbounded and monotonic (e.g., an increasing sequence unbounded from above diverges to $+\infty$).
3. It has a divergent subsequence.

Bolzano-Weierstrass Theorem (Implicit)

Every bounded sequence has a convergent subsequence.

Example: $u_n = \cos \frac{n\pi}{3}$ is bounded but divergent. However, $(u_{6n}) = \cos(2n\pi) = 1$ is a convergent subsequence.

Continuity and Limits of Sequences

If $(u_n)$ is a convergent sequence with $\lim u_n = l$ and $f$ is a function continuous at $l$, then $\lim f(u_n) = f(l)$.

Adjacent Sequences

Definition

Two sequences $(u_n)$ and $(v_n)$ are adjacent if:

1. $(u_n)$ is increasing.
2. $(v_n)$ is decreasing.
3. $\lim (v_n - u_n) = 0$.

Example: $u_n = 1 - \frac{1}{n}$ and $v_n = 1 + \frac{1}{n^2}$ are adjacent.

• $u_n$ is strictly increasing: u_{n+1} - u_n = \frac{1}{n(n+1)} > 0.
• $v_n$ is strictly decreasing: v_{n+1} - v_n = \frac{-2n-1}{n^2(n+1)^2} < 0.
• $\lim (v_n - u_n) = \lim (1 + \frac{1}{n^2} - (1 - \frac{1}{n})) = \lim (\frac{1}{n^2} + \frac{1}{n}) = 0$.

Theorems on Adjacent Sequences

1. If $(u_n)$ and $(v_n)$ are adjacent with $(u_n)$ increasing and $(v_n)$ decreasing, then $\forall n \in \mathbb{N}: u_n \leq v_n$.
   • Proof sketch: Let $w_n = v_n - u_n$. Since $u_n$ is increasing and $v_n$ is decreasing, $w_n$ is decreasing ($w_{n+1} - w_n \leq 0$). Since $\lim w_n = 0$, all $w_n$ must be $\geq 0$. Thus $v_n - u_n \geq 0 \Rightarrow u_n \leq v_n$.
2. Adjacent sequences are convergent and converge to the same limit $l$. Additionally, $\forall n \in \mathbb{N}: u_n \leq l \leq v_n$.
   • Proof sketch: $(u_n)$ is increasing and bounded above by $v_1$ (since $u_n \leq v_n \leq v_1$), so it converges to its supremum. $(v_n)$ is decreasing and bounded below by $u_1$ (since $u_1 \leq u_n \leq v_n$), so it converges to its infimum. Since $\lim (v_n - u_n) = 0$, their limits must be identical.

Mathematical Analysis: Real Sequences Cheatsheet

This cheatsheet provides a concise overview of real sequences, including definitions, properties, and important theorems for understanding their behavior.

1. Definition of a Real Sequence

• Definition: A sequence of real numbers is a function defined on a set of the form where is a non-negative integer.

• Notation:

  • denotes a sequence defined on .

  • is denoted by , which is called the general term.

• Types of Definitions:

  • Explicit: General term is defined directly.

    • Example: with .

      • , , .

  • Recurrent: Next term defined by previous terms.

    • Example: with , .

      • .

      • .

2. Bounded Sequences

• Definition: Let be a real sequence.

  1. Bounded Above: .

     • Example: is bounded above by $0<span data-latex=". (" data-type="inline-math"></span>u_n \leq 0$)

  2. Bounded Below: .

     • Example: is bounded below by $0<span data-latex=". (" data-type="inline-math"></span>u_n \geq 0$)

  3. Bounded: .

     • Alternative Definition: .

     • Example: is bounded since .

3. Monotonic Sequences (Variations)

• Definitions:

  1. Increasing: .

  2. Strictly Increasing: .

     • Example: ()

  3. Decreasing: .

  4. Strictly Decreasing: .

     • Example: ()

  5. Constant: .

• Monotonic: A sequence is monotonic if it is either increasing or decreasing.

• Studying Variations:

  1. Difference: Analyze the sign of .

     • If strictly increasing

     • If strictly decreasing

  2. Ratio (for positive terms): Compare .

     • If increasing

     • If $\frac{u_{n+1}}{u_n} \leq 1 \implies</p></li></ul></li></ol></li></ul><p></p>$ decreasing

• Theorem for Recurrent Sequences:

  Let $u_{n+1} = f(u_n)$ where $f$ is an increasing function.

  1. If $u_2 - u_1 \geq 0 \implies (u_n)$ is increasing.
  2. If $u_2 - u_1 \leq 0 \implies (u_n)$ is decreasing.

  • Example: $u_1 = -2$, $u_{n+1} = 3u_n + 1$. Here $f(x) = 3x + 1$ (increasing).
    • $u_2 = 3(-2) + 1 = -5$.
    • $u_2 - u_1 = -5 - (-2) = -7 \leq 0$.
    • Therefore, $(u_n)$ is decreasing.

4. Limits of Real Sequences

• Convergence (Finite Limit $l$):

  $(u_n)$ converges to $l$ if: $\forall \varepsilon > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow |u_n - l| \leq \varepsilon$.

  • Interpretation: All terms $u_n$ for $n \geq n_0$ fall within the interval $[l - \varepsilon, l + \varepsilon]$.
  • Notation: $\lim u_n = l$.
  • Example: To show $\lim \frac{n+1}{n+2} = 1$: $|u_n - 1| \leq \varepsilon \implies \frac{1}{n+1} \leq \varepsilon \implies n \geq \frac{1}{\varepsilon} - 1$. Choose $n_0 = \lceil \frac{1}{\varepsilon} \rceil$.
• Divergence: A sequence is divergent if it is not convergent.
  • This includes cases where $\lim u_n = \pm \infty$ or the limit does not exist (e.g., $u_n = (-1)^n$).
• Infinite Limit ($+\infty$):

  $\lim u_n = +\infty$ if: $\forall A > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow u_n \geq A$.

  • Example: $u_n = 2n + 1$. To show $\lim u_n = +\infty$: $2n+1 \geq A \implies n \geq \frac{A-1}{2}$. Choose $n_0 = \lfloor A \rfloor + 1$.
• Infinite Limit ($-\infty$):

  $\lim u_n = -\infty$ if: $\forall A > 0, \exists n_0 \in \mathbb{N}, \forall n \in \mathbb{N}: n \geq n_0 \Rightarrow u_n \leq -A$.

  • Example: $u_n = -3n + 2$. To show $\lim u_n = -\infty$: $-3n+2 \leq -A \implies n \geq \frac{A+2}{3}$. Choose $n_0 = \lfloor \frac{A+2}{3} \rfloor + 1$.

5. Properties of Limits

• Uniqueness of Limit: If a sequence converges, its limit is unique.
• Boundedness of Convergent Sequences: If $(u_n)$ converges, then $(u_n)$ is bounded.
  • Proof Idea: For sufficiently large $n$, $u_n$ is close to $l$, so it's bounded within $(l-\varepsilon, l+\varepsilon)$. The finite number of initial terms are also bounded, making the entire sequence bounded.
• Arithmetic of Limits: If $\lim u_n = l$ and $\lim v_n = l'$.
  1. $\lim (\lambda u_n) = \lambda l$ for $\lambda \in \mathbb{R}$.
  2. $\lim (u_n + v_n) = l + l'$.
  3. $\lim (u_n v_n) = l l'$.
  4. If $u_n \neq 0$ for $n \geq n_0$ and $l \neq 0$, then $\lim \frac{1}{u_n} = \frac{1}{l}$.
  5. If $v_n \neq 0$ for $n \geq n_0$ and $l' \neq 0$, then $\lim \frac{u_n}{v_n} = \frac{l}{l'}$.
• Absolute Value and Limit: If $\lim u_n = l \implies \lim |u_n| = |l|$.
  • Key Inequality: $||a| - |b|| \leq |a - b|$.
• Null Sequence: A sequence $(u_n)$ is a null sequence if $\lim u_n = 0$.
  • $\lim u_n = 0 \iff \lim |u_n| = 0$.
  • Example: $\lim \frac{(-1)^n \sqrt{n}}{n^2} = 0$.

6. Monotone Convergence Theorem

• Theorem:
  1. If $(u_n)$ is increasing and bounded above, it converges to its supremum.
  2. If $(u_n)$ is decreasing and bounded below, it converges to its infimum.
• Corollary: Every monotonic and bounded sequence is convergent.
  • Example Proof: $u_n = \sum_{k=1}^n \frac{1}{n+k}$.
    • $u_{n+1} - u_n = \frac{1}{(2n+1)(2n+2)} > 0 \implies (u_n)$ is strictly increasing.
    • $u_n = \sum_{k=1}^n \frac{1}{n+k} \leq \sum_{k=1}^n \frac{1}{n+1} = \frac{n}{n+1} < 1 \implies (u_n)$ is bounded above by 1.
    • Thus, $(u_n)$ converges.
• Application Example: $u_1 = 1$, $u_{n+1} = \frac{1+u_n^2}{2}$.
  1. Boundedness: $0 < u_n \leq 1$ for all $n$. (Proof by induction)
  2. Monotonicity: $u_{n+1} - u_n = \frac{(u_n-1)^2}{2} \geq 0 \implies (u_n)$ is increasing.
  3. Convergence: Since $(u_n)$ is increasing and bounded above, it converges.
  4. Limit: If $\lim u_n = l$, then $l = \frac{1+l^2}{2} \implies l^2 - 2l + 1 = 0 \implies (l-1)^2 = 0 \implies l = 1$.

7. Limits and Inequalities

• Theorem 1: If $\lim u_n = l$ and $u_n \geq 0$ for sufficiently large $n$, then $l \geq 0$.
• Theorem 2: If $\lim u_n = l$ and $u_n \leq 0$ for sufficiently large $n$, then $l \leq 0$.
• Theorem 3 (Limits preserve weak inequalities): If $a_n \to l$ and $b_n \to l'$, and $a_n \leq b_n$ for sufficiently large $n$, then $l \leq l'$.
  • Remark: Limits do not preserve strict inequalities. E.g., $\frac{1}{n^2} < \frac{1}{n}$, but $\lim \frac{1}{n^2} = \lim \frac{1}{n} = 0$.
• Theorem 4 (Infinite limits and inequalities):
  1. If $a_n \to +\infty$ and $a_n \leq b_n$ for sufficiently large $n$, then $b_n \to +\infty$.
     • Example: $\lim \frac{e^n + \sqrt{n^2 + 1}}{\sqrt{n + 1}} = +\infty$.
  2. If $b_n \to -\infty$ and $a_n \leq b_n$ for sufficiently large $n$, then $a_n \to -\infty$.
• Squeeze Theorem (Sandwich Theorem):

  If $u_n \leq w_n \leq v_n$ for sufficiently large $n$, and $\lim u_n = \lim v_n = l$, then $(w_n)$ is convergent and $\lim w_n = l$.

  • Example 1: $\lim \frac{n(\cos(n) + \sin(n))}{(n + 1)^2} = 0$.
    • Use bounds: $0 \leq |\frac{n(\cos(n) + \sin(n))}{(n + 1)^2}| \leq \frac{2n}{(n+1)^2}$.
    • Since $\lim \frac{2n}{(n+1)^2} = 0$, the limit is $0$.
  • Example 2: $\lim \frac{E(\sqrt{n}) + 1}{n^2} = 0$.
    • Use property: $\sqrt{n}-1 < E(\sqrt{n}) \leq \sqrt{n}$.
    • This implies $\frac{\sqrt{n}}{n^2} < \frac{E(\sqrt{n})+1}{n^2} \leq \frac{\sqrt{n}+1}{n^2}$.
    • Since $\lim \frac{\sqrt{n}}{n^2} = 0$ and $\lim \frac{\sqrt{n}+1}{n^2} = 0$, the limit is $0$.

8. Subsequences

• Definition: A subsequence of $(u_n)$ is formed by selecting terms $u_{k_n}$ where $(k_n)$ is a strictly increasing sequence of natural numbers.
  • Example: $(u_{2n})$, $(u_{2n+1})$, $(u_{n^2})$ are subsequences of $(u_n)$.
  • Important Property: If $(u_n)$ converges to $l$, then any subsequence $(u_{k_n})$ also converges to $l$.
• Divergence Criteria: A sequence $(u_n)$ is divergent if:
  1. It has two convergent subsequences with different limits.
     • Example: $u_n = (-1)^n$. $u_{2n} \to 1$, $u_{2n+1} \to -1$. Since $1 \neq -1$, $(u_n)$ is divergent.
  2. It is unbounded.
     • Theorem: An increasing sequence that is unbounded above diverges to $+\infty$.
  3. It admits a divergent subsequence.
• Bolzano-Weierstrass Theorem: Every bounded sequence has a convergent subsequence.
  • Example: $u_n = \cos \frac{n\pi}{3}$ is bounded and divergent, but $(u_{6n})_{n \in \mathbb{N}}$ converges to 1.

9. Continuous Functions and Limits

• Theorem: If $(u_n)$ converges to $l$ and $f$ is continuous at $l$, then $\lim f(u_n) = f(l)$.

10. Adjacent Sequences

• Definition: Two sequences $(u_n)$ and $(v_n)$ are adjacent if:
  1. $(u_n)$ is increasing.
  2. $(v_n)$ is decreasing.
  3. $\lim (u_n - v_n) = 0$.
• Property: If $(u_n)$ and $(v_n)$ are adjacent, then for all $n$, $u_n \leq v_n$.
• Theorem: If $(u_n)$ and $(v_n)$ are adjacent, then:
  1. They are both convergent and converge to the same limit $l$.
  2. For all $n$, $u_n \leq l \leq v_n$.