Problem 3. Show that for a sequence $(x_n)$ the following are true:
(i) $\lim x_n=0$ if and only if $\lim |x_n|=0$.
(ii) $\lim x_n=L$ implies $\lim |x_n|=|L|$. Is the converse true? Prove or give a counterexample.
(i) is already done, easy.
I'm halfway done with (ii), I split it into three cases:
$L = 0$, $L > 0$ and $L < 0$.
For $L = 0$ I just refer to part (i).
For $L > 0$, we know that $|x_n - L| < \epsilon$
if lim $|x_n|$ = $|L|$, then we must have
$||x_n| -|L|| < \epsilon$
but by the reverse triangle equality, $||x_n| - |L|| < |x_n - L|$ so clearly $||x_n|-|L|| < \epsilon$, thus
lim $x_n = L$ $\implies$ lim $|x_n| = |L|$ for $L > 0$
for some reason I'm confused as to part three, $L<0$. I'm having trouble seeing the difference between $L>0$ and $L<0$, and i'm starting to think that splitting it up like that isn't necessary at all.