Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ I stumbled on the following inequality: For all $n\geq 1,$
$$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}.$$
However I cannot find the proof of this anywhere.
Any ideas how to proceed?
Edit: I posted a follow-up question about generalizations of this inequality here:
Square root inequality revisited

As $\sqrt n>0,$
$$2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}\iff2\sqrt{n(n+1)}-2n<1$$
$$\iff2\sqrt{n(n+1)}<2n+1$$
Squaring we get
$$4(n^2+n)<(2n+1)^2=4n^2+4n+1\iff 1>0$$ which is true.
Can you follow the same method for the other inequality?
