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Definition: Sequence
A sequence is function whose domain is a set of the form $\set{ n \in \Z \mid n \geq m } $; $m =0$ or $m = 1$. We denote the sequence by a letter such as $s $ and denote its value at $n $ as $s_n $.
Definition: Limit
A sequence $(s_n) $ of real numbers is said to converge to the real number $s $ provided that
for each $\epsilon > 0$ there exists a number $N $ such that
$$n > N \to \abs{ s_n -s } < \epsilon $$
If $(s_n) $ converges to $s$, we will write $\limu{ n }{ \infty }s_n = s $, or $s_n \to s $.
The number $s $ is called the limit of the sequence $(s_n) $.
A sequence that does not converge to some real number is said to diverge.
Example
$s_n = \frac{ 1 }{ n }, \limu{ n }{ \infty } s_n = 0 $
Proof
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<span class="proof__expand"><a>[expand]</a></span>