$\{g_n\}$ be uniformly bounded sequence of functions on $[0,1]$ and converges pointwise to $g$.

Question

My question is -suppose $\{g_n\}$ be uniformly bounded sequence of functions on $[0,1]$ and converges pointwise to a function. Then can I say $\int_{0}^{1} |g_n(x)-g(x)| dx\to 0$ as $n\to\infty$?
If I can conclude $\{g_n\}$ converges uniformly to $g$, then I easily say that the above statement is true.
I first thought that uniform boundedness may imply uniform convergence but it is not true (example: $g_n(x)=x^n$ on $[0,1]$, it is uniformly bounded by $1$).
I can't prove or disprove the statements.
Can anybody answer the question? Thanks for assistance in advance.