Given that $x_n\geqslant 0$, $x_{n+1}\leqslant x_n+\dfrac{1}{n^2}$, prove this sequence converges.
I want to prove by The monotone bounded convergence theorem.
$$ 0\leqslant x_n=\displaystyle\sum_{i=1}^{n-1}(x_{i+1}-x_i)\leqslant\displaystyle\sum_{i=1}^{n-1}\dfrac{1}{i^2}<\dfrac{\pi^2}{6} $$ My question is how to prove the sequence is monotonous.
