So, you can use some standard results from analysis to deduce that, but one could argue they obscure just how simple this question is once you have the right point of view.
If there are infinitely many terms of the sequence in $[-M,M]$, then it's quite obvious there are infinitely many in either $[-M,0]$ or $[0,M]$. Say it's $[0,M]$ WLOG; then continuing there are infinitely many in either $[0,M/2]$ or $[M/2,M]$, and so on. This defines an infinite nested sequence of intervals that will converge to some number (the intersection of all the intervals), and this number is the limit of a convergent subsequence of the $x_i$ (take one term from each interval). So this shows the first part, and the second part now follows easily :)