Deduce that $\displaystyle 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{ 1}{\sqrt{n}}<2\sqrt{n}$
Since the question says deduce, I'm not sure if a solution that involves induction is acceptable.
Anyway, using induction,
We will need to prove: $\displaystyle 2\sqrt{n}+\frac{1}{\sqrt{n+1}}<2\sqrt{n+1}$
$\displaystyle \Leftrightarrow 2\sqrt{n}<\frac{2(n+1)-1}{\sqrt{n+1}}$
$\displaystyle \Leftrightarrow 2\sqrt{n}\sqrt{n+1}<{2n+1}$
$\displaystyle \Leftrightarrow 4n(n+1)<4n^2+4n+1$
$\displaystyle \Leftrightarrow 0<1$
which is true.