Convergence of a sequence of non-negative real numbers $x_n$ given that $x_{n+1} \leq x_n + 1/n^2$.

Note that $$ \lim_{n\to\infty}\sum_{k=n}^\infty\frac1{k^2}=0 $$ and for $m\gt n$, $$ a_m\le a_n+\sum_{k=n}^\infty\frac1{k^2} $$ First, take the $\limsup\limits_{m\to\infty}$: $$ \limsup_{m\to\infty}a_m\le a_n+\sum_{k=n}^\infty\frac1{k^2} $$ which must be non-negative. Then take the $\liminf\limits_{n\to\infty}$: $$ \limsup_{m\to\infty}a_m\le\liminf_{n\to\infty}a_n $$ Thus, the limit exists.

Your last equation has a couple of bugs in it: you should write $\limsup$ instead of $\lim$ because you haven't yet shown that the limit exists, and the index of summation on the righthand side should match the index in the other RHS term.

Once you fix it up, though, it'll give you what you want. Since $\sum \frac{1}{k^2}$ converges, you can make the quantity $(\limsup_n x_n) - x_k$ as small as you like by choosing $k$ large enough.