show that sequence of functions $f_n(x) = (nx^2+1)/(nx+1)$ converges uniformly on $[1,2]$

Question

I have found out that this sequence of functions approaches to $1$ at $x=1$, approaches to $2$ at $x=2$, approaches to $x$ when $1<x<2$. Now to prove the sequence converges uniformly in $[1,2]$, what should be N ?? And what is the trick to find N ?

Answer 1

If $x\in[1,2]$, then\begin{align}|f_n(x)-x|&=\left|\frac{nx^2+1}{nx+1}-x\right|\\&=\left|\frac{1-x}{nx+1}\right|\\&\leqslant\frac1n.\end{align}Can you take it from here?