Question 100
a. Prove that for every natural number $n$ it holds
$$
\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2 \sqrt{n}}\lt \sqrt{n}-\sqrt{n-1}
$$
b. Prove that an integer is part of an expression
$$
1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{m^{2}-1}}+\frac{1}{\sqrt{m^{2}}}
$$
where $m$ is a natural number equal to $2 m-2$ or $2 m-1$.
$$
\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2 \sqrt{n}}\lt \sqrt{n}-\sqrt{n-1}
$$
b. Prove that an integer is part of an expression
$$
1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{m^{2}-1}}+\frac{1}{\sqrt{m^{2}}}
$$
where $m$ is a natural number equal to $2 m-2$ or $2 m-1$.