Skip to main content

Sequence

∀n∈N\forall n\in \mathbb{N} we define the list number …,xn,xn+1,…\dots,x_n,x_{n+1},\dots is a sequence. We use (xn)(x_n) to denote; sometimes we have multiple variable and we may have something like (xa,b)a∈N(x_{a,b})_{a\in N} means this sequence focus on the changing of aa

  • Notice, sequence can be finite but subsequence can't. That's only infinite sequence have subsequence.

Limits for Sequence​

Let L∈RL\in \R, (an)n=1∞,an∈R(a_n)^{\infty}_{n=1}, a_n \in \R be a sequence.

  • ∀ϵ>0,∃N∈N s.t.∀n≥N,∥an−L∥<ϵ  ⟹  L\forall \epsilon > 0, \exists N\in \N \ s.t. \forall n \ge N, \|a_n - L\| < \epsilon \implies L is the limit of (an)n=1∞(a_n)^{\infty}_{n=1} denote as lim⁡n→∞an=L\lim\limits_{n\to \infty} a_n = L
  • We also say lim⁡n→∞an=L\lim\limits_{n\to \infty} a_n = L as (an)n=1∞(a_n)^{\infty}_{n=1} converges to LL

(UNIQUENESS OF LIMITS): lim⁡n→∞an=L  ⟺  lim⁡n→∞an=M  ⟹  L=M\lim\limits_{n\to \infty} a_n = L \iff \lim\limits_{n\to \infty} a_n = M \implies L = M. In other words, if a sequence converge to the two limits, then they are equal.

(SQUEEZE THEOREM) Suppose that there are three sequences (an),(bn),(cn)(a_n), (b_n), (c_n) satisfy an≤bn≤cn,∀n≥1a_n \le b_n \le c_n, \forall n \ge 1 and lim⁡n→∞an=lim⁡n→∞cn=L  ⟹  lim⁡n→∞bn=L\lim\limits_{n\to \infty} a_n = \lim\limits_{n\to \infty} c_n = L \implies \lim\limits_{n\to \infty} b_n = L.

(Theorem 2.5.2 Davidson. ): Let lim⁡n→∞an=L,lim⁡n→∞bn=M,α∈R\lim\limits_{n\to \infty} a_n = L, \lim\limits_{n\to \infty} b_n = M, \alpha \in \R

  1. lim⁡n→∞an+bn=L+M\lim\limits_{n\to \infty} a_n + b_n = L + M
  2. lim⁡n→∞αan=αL\lim\limits_{n\to \infty} \alpha a_n = \alpha L
  3. lim⁡n→∞anbn=LM\lim\limits_{n\to \infty} a_nb_n = LM
  4. lim⁡n→∞αan/bn=L/M,\lim\limits_{n\to \infty} \alpha a_n/b_n = L/M, if M≠0M\ne 0

lim sup⁡n→∞=lim⁡n→∞(sup⁡m≥nxm)\limsup\limits_{n\to\infty} = \lim\limits_{n\to \infty}(\sup\limits_{m\ge n} x_m) and lim inf⁡n→∞=lim⁡n→∞(inf⁡m≥nxm)\liminf\limits_{n\to\infty} = \lim\limits_{n\to \infty}(\inf\limits_{m\ge n} x_m)

Monotone Sequences​

(Theorem 2.6.1.): A monotone increasing sequence that is bounded above converges. A monotone decreasing sequence that is bounded below converges.

(Nested Intervals Lemma): Suppose that In=[an,bn]={x∈R:an≤x≤bn}I_n = [a_n, b_n]=\{x\in \R: a_n \le x \le b_n\} are nonempty closed intervals such that ∀n≥1,In+1⊆In\forall n \ge 1, I_{n+1} \subseteq I_n . Then the intersection ∩n≥1In\cap_{n\ge 1} I_n is nonempty.

Subsequences​

A subsequences of a sequence (a_n)∞_n=1(a\_n)^{\infty}\_{n=1} can be denote as (a_n_k)_k=1∞(a\_{n\_k})\_{k = 1}^{\infty} which also is a sequence.

(BOLZANO-WEIERSTRASS THEOREM): Every bounded sequence of real numbers has a convergent subsequence.

Cauchy Sequences​

(2.8.1. Proposition): Let (an)n=1∞(a_n)^{\infty}_{n=1} be a sequence converging to LL. ∀ϵ>0,∃N∈Z,s.t.∀m,n≥N,∥an−am∥<ϵ\forall \epsilon >0, \exists N\in \Z, s.t.\forall m,n \ge N, \|a_n - a_m\| < \epsilon

A sequence (an)n=1∞(a_n)^{\infty}_{n=1} of real numbers is called a Cauchy Sequence   ⟺  ∀ϵ>0,∃N∈Z,s.t.∀m,n≥N,\iff \forall \epsilon >0, \exists N\in \Z, s.t.\forall m,n \ge N, ∥an−am∥<ϵ\|a_n - a_m\| < \epsilon

  • Obviously, Cauchy sequence of real numbers converges.

(2.8.3. Proposition): Every Cauchy sequence is bounded

A subset S⊆RS\subseteq \R is complete   ⟺  ∀\iff \forall Cauchy sequence (an)∈S(a_n) \in S converges to a point in SS.

(Theorem 2.8.5, COMPLETENESS THEOREM): Every Cauchy sequence of real numbers converges so R\R is complete.