irrationality proof of $\sqrt{n}+\sqrt{n+1}$ for any $n>0$ irrationality proof of $\sqrt{n}+\sqrt{n+1}$ for any $n>0$
My attempt:
$\sqrt{n}+\sqrt{n+1}=\frac{p}{q}$
$2n+1+2\sqrt{n(n+1)}=\frac{p^2}{q^2}$
Now we have to show $2n+1+2\sqrt{n(n+1)}$ cannot be a perfect squere but I don't know how.

If $\sqrt{n+1}+\sqrt{n}$ is rational, so is
$$
\frac{1}{\sqrt{n+1}+\sqrt{n}}=\sqrt{n+1}-\sqrt{n}
$$
By summing and subtracting, we get that both $\sqrt{n+1}$ and $\sqrt{n}$ are rational.
It's not difficult to show that $\sqrt{n}$ is rational if and only if $n$ is a perfect square, but the only consecutive squares in the natural numbers are $0$ and $1$.
