<s>[INST] Is $\lim_{n\to\infty} \left( (\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} \right)$ convergent? 
$\lim_{n\to\infty} \left( (\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} \right)$ : convergent?

My attempt
$$
(\sum^n_{m=1} \frac{1}{\sqrt{m}}) - \sqrt{n} = (\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{n}}) + (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{n}}) + \cdots + (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n}})
$$
But I stuck here.
 [/INST] Hint:
Note that for $1\leq j \leq \frac{n}{2}$ we have
$$ \frac{1}{\sqrt{j}} - \frac{1}{\sqrt{n}} \geq \frac{\sqrt{2}}{\sqrt{n}}- \frac{1}{\sqrt{n}} \geq \frac{1}{10} \cdot \frac{1}{\sqrt{n}} $$
Use this to find a lower bound of your sequence which divergeces.
</s>