Prove $ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\dots+\frac{1}{(n+1)\sqrt{n}}<2$ 
For any positive integer $n$ prove by induction that:
  $$ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\dots+\frac{1}{(n+1)\sqrt{n}}<2.$$  

The author  says that   it  is  sufficient to prove  that
$$ \frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\dots+\frac{1}{(n+1)\sqrt{n}}<2-\frac{2}{\sqrt{n+1}}.$$
Why? Where this $\frac{2}{\sqrt{n+1}}$ term come from?

The idea of the author is that if you "stop" the sum at the $n$'th term you get this artificial bound of $2-$something. Then you can show by induction that this holds for every $n$. Having proven this, the assertion follows by taking the limit as $n\rightarrow \infty$ since the bound will always be less than $2$.
Long story short, the author creates a bound which is provable by induction, that's where the $\frac{2}{\sqrt{n+1}}$ comes from
