uniform convergence of $\frac{ x^2}{x^2+(1-nx)^2}$

Question

The sequence $$f_n=\frac{x^2}{x^2+(1-nx)^2}$$ is not uniformly convergent on $[0,1]$. At $x=0 \text{ and } 1 $ it converges to zero. Now consider $$M_n=\sup\{|f_n(x)-0|\}$$ but $M_n\geq1$ and therefore $M_n $ doesn't converge to 0. Given sequence is not uniformly convergent and also not equicontinuous.

Is the above approach correct?

Answer 1

Yes, your approach is correct. You should include proofs for

1. $f_n(x) \to 0$ for all $x \in [0,1]$

and

2. $M_n \ge 1$ for all $n$.