Let $F : [0, \infty) \rightarrow [0, \infty)$ be a monotone increasing function with $$ \lim_{x \to \infty} \frac{F(x)}{x} = +\infty $$ Prove that if $g_n : [0,1] \to [0,\infty)$ is a sequence of continuous functions satisfying $$\int_{0}^{1} F(|g_n'(x)|) \, dx + \int_{0}^{1} g_n(x) \,dx \leq M$$ for some $M>0$ then $g_n$ ha a uniformly convergent subsequence.
Any suggestions would be appreciated. One hint was given such a sequence of functions $f_n$, we need to show that there exists a function $\omega : [0,\infty) \to [0, \infty)$ with limit as $z \to 0$ of $\omega(z) = 0$, so that $$|g_n(x) - g_n(y)| \leq \omega(x-y)$$