I don't understand this one part in the proof for convergent sequences are bounded.
Proof:
Let $s_n$ be a convergent sequence, and let $\lim s_n = s$. Then taking $\epsilon = 1$ we have:
$n > N \implies |s_n – s| < 1$
From the triangle inequality we see that: $ n > N \implies|s_n| – |s| < 1 \iff |s_n| < |s| + 1$.
Define $M= \max\{|s|+1, |s_1|, |s_2|, …, |s_N|\}$. Then we have $|s_n| \leq M$ for all $n \in N$.
I do not understand the defining $M$ part. Why not just take $|s| + 1$ as the bound, since for $n > N \implies |s_n| < |s| + 1$?
Best Answer
$|s|+1$ is a bound for $a_n$ when $n > N$. We want a bound that applies to all $n \in \mathbb{N}$. To get this bound, we take the supremum of $|s|+1$ and all terms of $|a_n|$ when $n \le N$. Since the set we're taking the supremum of is finite, we're guaranteed to have a finite bound $M$.