ANA1_L7

Saw in $\mathbb{R}$:

• A monotonic increasing sequence bounded above converges to its supremum.
• ditto for monotonically decreasing sequences.

For a monotonic sequence $S$,
$S$ converges $⟺$ $S$ is bounded.
The $\Leftarrow$ proof is clear.
Proof for the $\Rightarrow$ direction:
Assume $S$ is monotonically increasing (same proof works for monotonically decreasing $S$).
Then, $S_1$ is a lower bound.
$S_n\to s$ then $s$ is upper bound
this is easy to check. Use definition of convergence.

Rudin, 3.2

defined open ball, neighborhood

$S\subset (X,d)$ is bounded if there exists $M$ such that $d(p,q)<M\forall p,q\in S$ $⟺$ $S$ is a subset of a ball.

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Cauchy sequenecs
Every cauchy sequence converges
Proof plan

• show cauchy imples bounded
• find monotonic subseq in cuchy subseq
  Every infinite bounded seq has a cnovergent subsequence.