Sequences

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:"",“label”:"",“_index”:0}] Definition 1. A sequence of real numbers is a set

$$a_1,a_2,a_3\dots$$

of real numbers indexed by the natural numbers. Usually, we write $(a_n{)}_{n=1}^{\infty }$ or $a_n$.

Examples §

$a_n=\frac{1}{n}$: we can see that $a_n\to 0$ as $n\to \infty$, We can say that $a_n$ is converging to the limit $0$.

$a_n=(-1{)}^n⟹(a_n{)}_{n=1}^{\infty }=(-1,1,-1,1,\dots )$. This just oscillates and doesn’t converge.

Now, if $a_n=\frac{(-1{)}^n}{n}=(-1,\frac{1}{2},-\frac{1}{3},\frac{1}{4},\dots$, this also converges to the limit $0$.

Convergence §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Convergence”,“label”:“convergence”,“_index”:1}] Definition 2 (Convergence). A sequence $(a_n{)}_{n=1}^{\infty }$ converges to the limit $A$ iff for every $\forall ϵ>0,\exists N=N(ϵ)\in \mathbb{N}:|a_n-A|<ϵ,n\ge N$. We write ${\lim }_{n\to \infty }{a}_n=A$.

Examples §

Show that $a_n=\frac{1}{n}\to 0$.

\begin{proof} Fix $ϵ>0$ and choose $N>\dots$.

Then, $\forall n\ge N$, we have

$$|\frac{1}{n}-0|=\frac{1}{n}\le \frac{1}{N}<ϵ$$

\end{proof}

[!math|{“type”:“lemma”,“number”:“auto”,“setAsNoteMathLink”:false,“_index”:2}] Lemma 3. If $(a_n)$ and $(b_n)$ are sequences with ${\lim }_{n\to \infty }{a}_n=A$ and ${\lim }_{n\to \infty }{b}_n=B$, then a. ${\lim }_{n\to \infty }(a_n+b_n)=A+B$ \begin{proof}\Since ${\lim }_{n\to \infty }{a}_n=A$, $\exists N\mid |a_{n_{1A|<}}-\frac{ϵ}{2}\forall n_1\ge N_1$. If Since ${\lim }_{n\to \infty }{b}_n=B$, $\exists N_2\mid |b_n-B|<\frac{ϵ}{2},\forall n_2\ge N_2$. If we let $N=max(N_1,N_2)$, then $\forall n\ge N$, $|a_n+b_n-(A+B)|=|(a_n-A)+(b_n-B)$ $\le |a_n-A|+|b_n-B|<\frac{ϵ}{2}+\frac{ϵ}{2}=ϵ$. \end{proof} b. ${\lim }_{n\to \infty }c{a}_n=cA$ for $c\in \mathbb{R}$ c. ${\lim }_{n\to \infty }\frac{a_n}{b_n}$ d. ${\lim }_{n\to \infty }{a}_n{b}_n=AB$.

Monotone Sequence §

[!math|{“type”:“definition”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Monotonically Increasing Sequence”,“label”:“monotonically-increasing-sequence”,“_index”:3}] Definition 4 (Monotonically Increasing Sequence). A sequence $(a_n{)}_{n=1}^{\infty }$ is monotone increasing, written $a_n\uparrow$, if $a_n\le a_{n+1},\forall n\ge 1$ and is monotone decreasing, written $a_n\downarrow$ if $a_{n+1}\le a_n,\forall n\ge 1$. In either case, $a_n$ is monotone

[!math|{“type”:“theorem”,“number”:“auto”,“setAsNoteMathLink”:false,“title”:“Monotone Sequence Convergence”,“label”:“monotone-sequence-convergence”,“_index”:4}] Theorem 5 (Monotone Sequence Convergence). Every bounded monotone sequence is convergent. \begin{proof} By the ^e56e85 , $\{a_n\}$ has a supremum $A$. Then, for any $ϵ>0$, $A-ϵ$ is not an uppre bound, so $\exists N\mid a_n>A-ϵ$. But $(a_n)$ is monotone increasing, so $a_n\ge a_N\ge A-ϵ,\forall n\ge N$. \end{proof}

The Squeeze Principle §

If $(a_n)$, $(b_n)$, and $(c_n)$ are sequences such that

• $b_n\le a_n\le c_n,\forall n\ge i$
• ${\lim }_{n\to \infty }{b}_n={\lim }_{n\to \infty }{c}_n=L$

then ${\lim }_{n\to \infty }{a}_n=L$.

\begin{proof} Choose $N_1$ \end{proof}