I can prove the contrapositive:
$x_n$ does not tend to $0$ implies either:
- $x_n$ diverges (does not converge), in which case neither does $\|x_n\|$, or
- $x_n$ converges to $x \neq 0$ which implies $\|x_n\|$ converges to $\|x\|\neq 0$.
Okay, but is there a simpler way to do this?