Ex 3.1 of Baby Rudin

Question

I was trying to prove ex 3.1 by conventional method, using definition of limit of sequence. I tried very hard to prove the convergence of {$|S_n|$} under the assumption of {$|S_n|$} converge to same value as {$S_n$}. later I gave up and look the solution. It took entirely different approach,{$S_n$} is cauchy implies {$|S_n|$} is cauchy hence convergence and used norm distance function. Question: (1) can we show sequence $S_n$ and $|S_n|$ converge to same value, if {$S_n$} convergent. (2) can we prove ex 3.1 using generic metric? I think it is implicitly assume that {$S_n$} is in $\mathbb{R}$ or $\mathbb{R^k}$.

Answer 1

Notice that for any real numbers $x,y$, we have $$||x| - |y|| \leq |x-y|.$$ Let $s = \lim\limits_{n \to \infty} s_n$ and let $\epsilon > 0$ be given. We can then find $N \in \mathbb{N}$ so that for all $n \geq N$, $|s_n - s| < \epsilon$, so for $n > N$, we have $$||s_n| - |s|| \leq |s_n - s| < \epsilon,$$ so $(|s_n|) \to |s|$.