First, its partial sum is monotonically increasing because 

$$S_n - S_{n-1}=\frac{1}{n^2} > 0 $$

Next, let's show $S_n < 2$.

$$\begin{align*} S_n =\ &\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\\ <\ &\frac{1}{1^2}+ \frac{1}{1\times 2} + \frac{1}{2\times 3} +\cdots +\frac{1}{(n-1)\times n}\\ =\ &1 + \left(\frac{1}{1}-\frac{1}{2}\right)+ \left(\frac{1}{2}-\frac{1}{3}\right)+ \cdots + \left(\frac{1}{n-1}-\frac{1}{n}\right)\\ =\ & 2 - \frac{1}{n}\\ <\ & 2\end{align*}$$

Therefore, we conclude that this sequence is convergent to a limit less than $2$