Proving $\sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n}$ with induction

Question

Problem: Prove with induction that \begin{align*} \sum_{j=1}^n \frac{1}{\sqrt{j}} > \sqrt{n} \end{align*} for every natural number $n \geq 2$.

Attempt at proof: Basic step: For $n = 2$ we have $1 + \frac{1}{\sqrt{2}} > \sqrt{2}$ which is correct.

Induction step: Suppose the assertion holds for some natural number $n$, with $n > 2$. Then we now want to prove that it also holds for $n +1$, i.e. that \begin{align*} \sum_{j=1}^{n+1} \frac{1}{\sqrt{j}} > \sqrt{n+1} \end{align*} Now we have that \begin{align*} \sum_{j=1}^{n+1} \frac{1}{\sqrt{j}} = \sum_{j=1}^n \frac{1}{\sqrt{j}} + \frac{1}{\sqrt{n+1}} > \sqrt{n} + \frac{1}{\sqrt{n+1}} \end{align*} or \begin{align*} \sum_{j=1}^n \frac{1}{\sqrt{j}} + \frac{1}{\sqrt{n+1}} > \frac{\sqrt{n} \sqrt{n+1} + 1}{\sqrt{n+1}} \end{align*}

Now I'm stuck, and I don't know how to get $\sqrt{n+1}$ on the right hand side. Help would be appreciated.

Answer 1

As Daniel Fischer points out in the comments, since you have $$ \sum_{j=1}^{n+1} \frac{1}{\sqrt{j}} > \sqrt{n} + \frac{1}{\sqrt{n+1}} $$ it is enough to show $$ \sqrt{n} + \frac{1}{\sqrt{n+1}} \geq \sqrt{n+1} $$ or equivalently $ \frac{1}{\sqrt{n+1}} \geq \sqrt{n+1} - \sqrt{n} $. A way to show this final inequality is to recall the identity $(a-b)(a+b) = a^2-b^2$ and multiply both sides by $\sqrt{n+1} + \sqrt{n}$, i.e. to use the identity with $a=\sqrt{n+1}$ and $b=\sqrt{n}$: what happens to the LHS? To the RHS?

Answer 2

Notice that :

$ \sqrt{n}\sqrt{n+1} + 1 > \sqrt{n}\sqrt{n} + 1 $

$\sqrt{n}\sqrt{n+1} + 1 > n + 1 $

$ \sqrt{n}\sqrt{n+1} + 1 > \sqrt{n+1}\sqrt{n+1} $

$ \frac{\sqrt{n}\sqrt{n+1} + 1}{\sqrt{n+1}} > \sqrt{n+1}. $