Limit of a Sequence

Topics: Calculus - Sequence - Limit

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(definition)

Let $\{a_n{\}}_{n\in \mathbb{N}}\subset \mathbb{R}$ be a sequence and $L\in \mathbb{R}$.

We say that $L$ is the limit of $\{a_n{\}}_{n\in \mathbb{N}}$ when $n$ tends to infinity if $\forall \epsilon >0$, $\exists N\in \mathbb{N}$ such that $|a_m-L|<\epsilon$ if $m>N$.

This is denoted by:

$$\underset{n\to \infty }{\lim }{a}_n=L$$

Proof

Let $\epsilon >0$.

The Archimedian Property tells us that $\exists N\in \mathbb{N}$ such that $\frac{1}{N}<\epsilon$.

With that, if $m>N$, then:

$$\frac{1}{m}<\frac{1}{N}$$

Finally, we can see that:

$$\left|\frac{1}{m}-0\right|<\left|\frac{1}{N}-0\right|<\epsilon$$

Definition in terms of an open ball

We say that $L$ is the limit of $\{a_k\}$ if $\forall \epsilon >0$, $\exists N\in \mathbb{N}$ such that $a_m\in B_{\epsilon }(L)$ $\forall m>N$.

This is the same definition as above, but given in terms of an open ball.

When a sequence has a limit, we say that it converges. When it doesn’t, then we say it diverges.

Limits of sequences can be operated.

Limit of a Sum of Sequences §

(theorem)

Let $\{a_n\},\{b_n\}\subseteq \mathbb{R}$ be sequences such that ${\lim }_{n\to \infty }{a}_n=A$ and ${\lim }_{n\to \infty }{b}_n=B$.

Then, the sequence $\{c_n\}=\{a_n+b_n\}$ converges and ${\lim }_{n\to \infty }{c}_n=A+B$.

Limit of a Sequence Greater than 0 §

(theorem)

If $a_n\ge 0$ $\forall n\in \mathbb{N}$ and $\{a_n{\}}_{n\in \mathbb{N}}$ is convergent, then:

$$\underset{n\to \infty }{\lim }{a}_n\ge 0$$

Sandwich Limit §

(theorem)

If $\{a_n{\}}_{n\in \mathbb{N}}$, $\{b_n{\}}_{n\in \mathbb{N}}$, $\{c_n{\}}_{n\in \mathbb{N}}$ are convergent sequences such that:

$$a_n\le b_n\le c_n\forall n\in \mathbb{N}$$

Then:

$$\underset{n\to \infty }{\lim }{a}_n\le \underset{n\to \infty }{\lim }{b}_n\le \underset{n\to \infty }{\lim }{c}_n$$

(we can prove this theorem with the previous theroem)