The easy way is to show that $ a_{n} = - 2\sqrt {n+1} + ({\frac{1}{\sqrt {1}}+{\frac{1}{\sqrt{2}}}+\ldots+{\frac{1}{\sqrt{n}}}}) $ begins a little low and increases with $n,$ while
$ b_{n} = - 2\sqrt {n} + ({\frac{1}{\sqrt {1}}+{\frac{1}{\sqrt{2}}}+\ldots+{\frac{1}{\sqrt{n}}}}) $ begins a little high and decreases with $n,$
finally $b_n > a_n$ and $b_n - a_n \rightarrow 0.$
The conclusion is that all the $b_j$ are larger than all the $a_i,$ so the $a_n$ is an increasing sequence with an upper bound.
Worth actually calculating the two and printing side by side for, say, $n \leq 50.$ I did that, it is not until $n \geq 159$ that I finally get $ -1.5 < a_n < b_n < -1.4, $ as in
159 -1.499903567256619 -1.420722711746567