Proof: $2\sqrt{m}-2 < \sum\limits_{n=1}^m\frac{1}{\sqrt{n}}< 2\sqrt{m}-1$ I know that problems similar to this one, involving either one of the two bounds, have been posted before, but I would like just a hint in the last part of the proof involving the upper bound, with which I have some troubles with being ''convincing'' enough. 
Prove that $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$ if $n\geq 1$. Then use this to prove that 
$$2\sqrt{m}-2 < \sum_{n=1}^m\frac{1}{\sqrt{n}}< 2\sqrt{m}-1 \qquad \text{if }\ m \geq 2$$
With some help I received here I got the following proof for the first set of inequalities:
Proof (direct):
Since
$$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}} < \frac{1}{\sqrt{n} + \sqrt{n}} = \frac{1}{2\sqrt{n}}$$
then we have
$$2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}}\qquad (1)$$
Likewise, since
$$\frac{1}{2\sqrt{n}} = \frac{1}{\sqrt{n} + \sqrt{n}} < \frac{1}{\sqrt{n} + \sqrt{n-1}} = \sqrt{n} - \sqrt{n-1}$$
then
$$\frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})\qquad (2)$$
as asserted.
For the second part I tried to prove the lower bound firt:
Proof (direct): From $(1)$ we have
$$\sqrt{n+1} - \sqrt{n} < \frac{1}{2\sqrt{n}}$$
By summing both sides we get
$$\sum_{n=1}^{m}(\sqrt{n+1} - \sqrt{n}) < \sum_{n=1}^{m}\frac{1}{2\sqrt{n}}$$
And after applying the telescoping property on the LHS, this becomes
$$\sqrt{m+1} - 1 < \sum_{n=1}^{m}\frac{1}{2\sqrt{n}}$$
But since $\sqrt{m} - 1 < \sqrt{m+1} - 1$, by transitivity we have
$$\begin{align*}\sqrt{m} - 1 &< \sum_{n=1}^{m}\frac{1}{2\sqrt{n}}\\
2\sqrt{m} - 2 &< \sum_{n=1}^{m}\frac{1}{\sqrt{n}}\end{align*}$$
Then the upper bound:
Proof (direct): From $(2)$ we have
$$\frac{1}{2\sqrt{n}}< \sqrt{n} - \sqrt{n-1}$$
By taking the sum of both sides we get
$$\sum_{n=1}^{m}\frac{1}{2\sqrt{n}}< \sum_{n=1}^{m}(\sqrt{n} - \sqrt{n-1})$$
If we apply the telescoping property on the RHS, the inequality becomes
$$\begin{align*}\sum_{n=1}^{m}\frac{1}{2\sqrt{n}} &< \sqrt{n}\\
\sum_{n=1}^{m}\frac{1}{\sqrt{n}} &< 2\sqrt{n}\end{align*}$$
Here I'm not sure how to get the $-1$ on the RHS that I'm lacking. I could have started the summation at $n=2$ I guess, but don't think it's correct and it doesn't seem to be a convincing way to complete the proof.

Well you tagged induction so why not use it:
$$
\sum_{n=1}^{m+1} \frac{1}{\sqrt{n}} \lt 2\sqrt{m}-1 + \frac{1}{\sqrt{m+1}} \lt 2\sqrt{m}-1 + 2(\sqrt{m+1}-\sqrt{m}) = 2\sqrt{m+1}-1
$$
