Using properties of limits, calculate $\lim_{n\to \infty}\left(\frac1{n^2}+\frac1{(n+1)^2}+\cdots+\frac1{(2n)^2}\right)$ Using the properties of limits, calculate the following limits, if they exist. If not, prove they do not exist:
$$\lim_{n\to \infty}\left(\frac1{n^2}+\frac1{(n+1)^2}+\frac1{(n+2)^2}+\cdots+\frac1{(2n)^2}\right)$$
This is what I have done, I have expressed the limit in the form:
$\lim_{n\to \infty}\frac1{(n+a)^2}$ where 'a' belongs to the reals. 
Then using the $\epsilon-N$ definition of limits, I assumed that:
$$\lim_{n\to \infty}\frac1{(n+a)^2}=0$$ and carried forward with the proof. I would like to use the $\epsilon-N$ definition of limits since it is what we are covering right now, is this the right way of solving this problem?

Let F(n): $$\frac{1}{n^2}+\frac{1}{(n+1)^2}+\cdots+\frac{1}{(2n)^2}$$
Consider two series:
 $$G(n): \frac{1}{n^2}+\frac{1}{n^2}+\frac{1}{n^2}+\cdots +\frac{1}{n^2}$$
and 
$$H(n):\frac{1}{(2n)^2}+\frac{1}{(2n)^2}+\frac{1}{(2n)^2}+\cdots+\frac{1}{(2n)^2}$$
Notice that
$$H(n)<F(n)<G(n)$$
Now, $$\lim_{n \to \infty} G(n) = \lim_{n \to \infty} H(n)=0$$
Hence, by the sandwich theorem, the given limit becomes $0$
