Question

show that there is no positive integer n for which √n-1 +√n+1 is rational

Answer 1

There many positive integers for which √n-1 + √n +1 is rational.

For example take n=4

then √4 - 1 + √4 +1 = 2-1+2+1 = 4 which is rational. 

If your question is different pls post it again correctly.

Answer 2

if your question means
$\sqrt{n + 1} + \sqrt{n - 1}$