I'm not sure if this is true or not. I find it very intuitive:
$\bigcap_{k=1}^{\infty} [n,n+\frac{1}{n^{2}}+\frac{1}{k}) = [n,n+\frac{1}{n^{2}}] $
Is this true and how do I then construct a proof? If the case is that the above statement is false, are there any other ways to express $[n,n+\frac{1}{n^{2}}]$?