Proof of a sequence $\{S_n\}$ converging to $0$ First, the question asks

Prove that $1+\frac12+\frac13+\cdots+\frac1n<2\sqrt n$

At first glance, I see that this is a proof involving induction, and I arrived at an inequality as shown below where I complete my proof: 

$$\frac1{\sqrt{k+1}-\sqrt k}<2k+2$$

Then, the question asks further 

Let $S_n=\frac{1}{n}+\frac{1}{2n}+\frac1{3n}+\cdots+\frac1{n^2}$ for each $n\in\Bbb N$. Prove that $\{S_n\}$ converges to $0.$ 

The second part of the question is where I am stuck. I don't know how I can make use of the first part of the question to answer the latter part.

\begin{align}S_n&=\frac1n+\frac1{2n}+\frac1{3n}+\cdots+\frac1{n^2}\\[10pt]
&=\frac1n\left(1+\frac12+\frac13+\cdots+\frac1n\right)\\[10pt]
&\leqslant\frac{2\sqrt n}n\\[10pt]
&=\frac2{\sqrt n}.\end{align}
