sequence

Sequence

A sequence is an ordered “collection” of elements with indices subset of the natural numbers.
They may be finite (→ n-tuple) or infinite.

Formally, a sequence is an indexed family with the linearly ordered index set $\{n_0,\dots ,n\}\subseteq \mathbb{Z}$

$$x:\mathbb{N}\to X,n\to x_n$$

Sequences are commonly denoted as $(a_1,a_2,a_3,\dots ,a_n)$ or $(a_n{)}_{n=1}^{\infty }=(a_n{)}_{n\ge 1}=(a_k{)}_{k\in \mathbb{N}}$, or in context also $(a_n)$.
The starting index $n_0$ may vary, but is often $1$ or $0$.

The range of a sequence $(a_n{)}_{n\in I}$ is the set of values it takes: $\{a_n:n\in I\}$, same as the image of the function $a:I\to X$.

Ordered $⟹$ $(a,b)\ne (b,a)$, as opposed to sets $\{a,b\}=\{b,a\}$

Convergence

We say that the sequence $(a_n{)}_{n\in \mathbb{N}}\subseteq \mathbb{C}$ converges to $a$ iff:

$$\begin{array}{rl}& \forall \epsilon >0\exists n_0\in \mathbb{N}\forall n\ge n_0:|a_n-a|<\epsilon \\ ⟺& \forall ϵ>0\exists n_0\in \mathbb{N}\forall n\ge n_0:a_n\in U_{\epsilon }(a)\\ ⟺& \forall \epsilon >0\exists n_0\in \mathbb{N}:(a_n{)}_{n=n_0}^{\infty }\subseteq U_{\epsilon }(a)\end{array}$$

where $U_{\epsilon }(a)$ is the epsilon neighbourhood of $a$.
We then call $a$ the limit of the sequence and write:

$$\underset{n\to \infty }{\lim }{a}_n=a\text{or simply}{a}_n\to a$$

The sequence is called convergent, or we say that its limit exists, if such an $a$ exists. Otherwise, the sequence is called divergent.

Link to original

Null sequence

Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence such that it converges to 0:

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

Then we call $(a_n{)}_{n\in \mathbb{N}}$ a null sequence.

Link to original

Bounded sequences

Bounded sequences

Let $(a_n{)}_{n\in \mathbb{N}}\subseteq \mathbb{C}$ be a sequence. We call the sequence bounded iff

$$\exists R>0:\forall n\in \mathbb{N}:|a_n|\le R$$

For $(a_n{)}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ (why only $\mathbb{R}$?), we call the sequence bounded from above iff

$$\exists C\in \mathbb{R}:\forall n\in \mathbb{N}:a_n\le C$$

and bounded from below iff

$$\exists c\in \mathbb{R}:\forall n\in \mathbb{N}:a_n\ge c$$

In other words, a sequence is bounded (from above / below) if its range is a bounded set (from above / below)

EXAMPLE

$((-1{)}^n{)}_{n\in \mathbb{N}}$ is bounded by 1
${\left(\frac{42}{n}\right)}_{n\in \mathbb{N}}$ is bounded by 42
$((-1{)}^n\cdot \frac{42}{n}{)}_{n\in \mathbb{N}}$ is bounded by 42

Bounded sequences are closed under $+$ and $\cdot$

If $(a_n)$ is bounded by $R_1$ and $(b_n)$ by $R_2$:

• Sum: $|a_n+b_n|\le |a_n|+|b_n|\le R_1+R_2$ (triangle inequality)
• Product: $|a_n\cdot b_n|=|a_n|\cdot |b_n|\le R_1\cdot R_2$

A convergent sequence is bounded.

But the converse is not true: A bounded sequence need not be convergent (e.g. $((-1{)}^n)$).

Link to original