I have a question about the proof of Cauchy implying boundness. The proof argues that after $N$ terms, the sequence is bounded, i.e. we could have $|s_n - s_N| < 1, \forall n > N \Rightarrow s_n \in (s_N - 1, s_N + 1)$.
But to argue $\{s_n : n < N\}$ is bounded, the proof simply says that because it is finite, namely we can have $ |s_n| \leq \max\{ |s_1|, \ldots, |s_N| \}$. I don't quite understand this part. What if $s_{i} = \infty, i<N$? Then the sequence is not bounded.
Am I missing something obvious?
The author of that proof is just saying that if you have a finite set $A$ and if $a\in A$, then $|a|\leqslant\max\{|x|\mid x\in A\}$. Note that, since $A$ is a finite set of numbers (in particular, none of them is $\infty$), it has a maximum. And then each of its elements is smaller than or equal to that maximum.