Just a revision

We have,

$\Rightarrow \sqrt{n+1}-\sqrt{n-1}<0.2$

$\Rightarrow \sqrt{n+1}<0.2+\sqrt{n-1}$

Squaring both sides:

$\Rightarrow (\sqrt{n+1})^{2}<(0.2+\sqrt{n-1})^{2}$

$\Rightarrow n+1<0.04+n-1+0.4\sqrt{n-1}$

$\Rightarrow n+1-n+1-0.04<0.4\sqrt{n-1}$

$\Rightarrow \frac{1.96}{0.04}<\sqrt{n-2}$

$\Rightarrow 4.9<\sqrt{n-1}$

Squaring both sides:

$\Rightarrow (4.9)^{2}<(\sqrt{n-1})^{2}$

$\Rightarrow 24.01+1<n$

$\Rightarrow 25.01<n$

Therefore , $n=\color{#3D99F6}{\boxed{26}}$