Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.
$$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$
Perhaps someone can help me out I don't understand how to move forward from here:
$$\sum\limits_{k=1}^{n+1}{\frac{1}{\sqrt{k}}+\frac{1}{\sqrt{n+1}}\ge \sqrt{n+1}}$$
proof and explanation would greatly be appreciated :)
Thanks :)
EDIT sorry meant GE not = fixed :)

If you wanted to prove that
$$
\sum_{k=1}^n \frac 1{\sqrt k} \ge \sqrt n,
$$
that I can do. It is clear for $n=1$ (since we have equality then), so that it suffices to verify that
$$
\sum_{k=1}^{n+1} \frac 1{\sqrt k} \ge \sqrt{n+1}
$$
but this is equivalent to
$$
\sum_{k=1}^{n} \frac 1{\sqrt k} + \frac 1{\sqrt{n+1}} \ge \sqrt{n+1} \
$$
and again equivalent to
$$
\sum_{k=1}^n \frac{\sqrt{n+1}}{\sqrt k} + 1 \ge n+1
$$
so we only need to prove the last statement now, using induction hypothesis. Since
$$
\sum_{k=1}^n \frac 1{\sqrt k} \ge \sqrt n, 
$$
we have 
$$
\sum_{k=1}^n \frac{\sqrt{n+1}}{\sqrt k} \ge \sqrt{n+1}\sqrt{n} \ge \sqrt{n} \sqrt{n} = n.
$$
Adding the $1$'s on both sides we get what we wanted.
Hope that helps,
