If $\int_{0}^{1}|f_n(x)-f(x)|dx\to 0$ then $f_n\to f$ uniformly on $[0,1]$: True/false

Question

Let $\{f_n\}$ be a sequence of continuous real valued functions on $[0, \infty)$. Suppose $f_n(x)\to f(x) ~~~\forall x\in [0,\infty)$ and $f$ is integrable. Then

if $\int_{0}^{1}|f_n(x)-f(x)|dx\to 0$ then $f_n\to f$ uniformly on $[0,1]$: True/false

Im not able to find a counter example ?

Pliz help me

Answer 1

False .. $x^n$ is not uniformly convergent on $[0,1]$. Here the limit function is $f(x) = 0$ when $x \in [0,1)$ and $f(1) = 1$

But $\int_{0}^{1}|x^n-f(x)|dx\to 0$