Proving uniform continuity for a function given integral condition for its derivative

Question

I have a problem stating: Let $f$ be a continuously differentiable function in $[0,\infty)$, and suppose that $\int_0^{\infty}{\lvert f'(t)\rvert^2dt}<\infty$. Now I would like to show that f is uniformly continuous in $[0,\infty)$. My approach is that the integral-condition gives that there exists a constant $M>0$ such that $\lvert f'(t)\rvert<M$ for all $t>0$. From this then it is easy to show that $f$ is uniformly continuous. Am I correct or does anyone have a better idea?

Answer 1

For $x<y$ apply Holder's inequality to get $|f(x)-f(y)|=|\int_x^{y}f'(t)dt|\leq \sqrt {|x-y|}\sqrt {\int_x^{y}(f'(t))^{2}}dt$. Can you finish?