I am studying for an entrance exam in August and I am trying to review and cover the real analysis section. I came across this problem on one of the past exams and I am having some trouble with it.
Let $G(x, y)$ be a continuous function on $R^2$ and suppose for each positive integer $k$, that $g_k$ is a continuous function defined on [0, 1] with the property that $|g_k(y)| \leq 1$ for all $y \in [0, 1]$. Now define
$$f_k(x) := \int_{0}^{1}g_k(y)G(x, y) dy.$$ Prove that the sequence ${f_k}$ is equicontinuous on [0, 1].
So for the problem above, given that $[0,1]$ is compact, I think that it is sufficient to show that {${f_k(x)}$} is uniformly convergent on the interval $[0,1]$. How would I show that {${f_k(x)}$} is uniformly convergent on {${f_k(x)}$}?
You cannot prove convergence. Instead, take $\epsilon >0$ and choose $\delta >0$ such that $|G(x,y)-G(x',y')| <\epsilon$ whenever $\|(x,y)-(x',y')\| <\delta$. This is possible because any continuous function on the compact space $[0,1]\times [0,1]$ is unifomly continuous. From this you get $|f_k(x)-f_k(y)| <\epsilon$ whenever $|x-y| <\delta$ which is the definition of equicontinuity.