Uniform convergence in $[0,1]$

Question

I have problems with exercise:

Study the uniform convergence on $[0, 1]$ of the sequence $f_n$ defined by:

$f_n = \displaystyle\frac{t^2}{t^2+(nt-1)^2}$

My attempt:

$\displaystyle\lim_{n \to{}\infty}{ \displaystyle\frac{t^2}{t^2+(nt-1)^2}} = 0, f(t) = 0 $ for all $t\in [0,1]$

How can I check that $\sup_{t \in [0,1]}|f_n(t)-f(t)| = 0$ ?

Thanks

Answer 1

Your $f_n$ does not converge uniformly. Let $\varepsilon = \frac{1}{2}$.

Then for every $n$, $$\left|f_n\left(\frac{1}{n}\right) - 0\right| = 1 > \varepsilon$$