Applying the monotone convergence theorem

Question

Recently learned about the monotone convergence theorem.

I have the sequence: $x_n = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}$

I need help proving that it is increasing and bounded, hence converges by the monotone convergence theorem.

I can see that it is monotonically increasing and it's bounded below by 1 but I'm not sure how to prove these properties. Also how do I determine and prove the upper bound for the series?

Thanks in advance.

Answer 1

$\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}<\frac{1}{1*2}+\frac{1}{2*3}+...+\frac{1}{(n-1)*n}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}<1 $, so $ x_n<2 $ for all $ n $.