Berkeley exam summer '79, sequence of continuous functions, integral, convergence I've recently been browsing some Berkeley exams and I'm particularly interested in Problem 19 here.

Let ${f_n}$ be a sequence of continuous real functions deﬁned on $[0,1]$ such that $\int_0^1 (f_n(y))^2 dy \le 5$ 
  for all $n$. 
Deﬁne $g_n : [0,1] \rightarrow \mathbb{R}$ by
  $g_n(x) = \int ^1_0 \sqrt{x + y} f_{n} (y)dy$.
  
  
*
  
*Find a constant $K > 0$ such that $|g_n(x)| \le K$ for all $n$.
  
*Prove that a subsequence of the sequence $\{g_n\}$ converges uniformly.
  

Could you help me solve it?
Frankly speaking, I don't know how to use the condition that $\int_0^1 (f_n(y))^2 dy \le 5$.
 A: *

*By the Cauchy-Schwarz Inequality we have
$$
    \left|g_n(x)\right| \leqslant \sqrt{\int_0^1 (x+y) \, dy }
    \sqrt{\int_0^1 (f_n(y))^2 \, dy } \leqslant \sqrt{\int_0^1 (1+y) \, dy }\,
    \sqrt{5} = \sqrt{\frac{15}{2}}
$$


Alternatively, we can also use the fact that $\left|f_n (x)\right| \leqslant 1 + (f_n(x))^2$ and see that
$$\int_0^1 \left|f_n(x)\right| \, dx \leqslant 1 + 5 = 6,$$
so
$$\left|g_n(x)\right| \leqslant \sqrt{2} \int_0^1 \left|f_n(y)\right| dy \leqslant 6\sqrt{2}.$$


*

*Since $\sqrt{x+y}$ is a continuous function of $x$ and $y$ on the
compact unit square, it is uniformly continuous there. Hence, given
any $\varepsilon > 0$, there exists $\delta > 0$ such that
$$\left|\sqrt{x_1 + y_1} - \sqrt{x_2 + y_2}\right| < \varepsilon $$
whenever $\left|x_1 - x_2\right| + \left|y_1 - y_2 \right| < \delta$.
In particular, $\left|\sqrt{x_1 + y} - \sqrt{x_2 + y}\right| < \varepsilon$ whenever
$\left|x_1 - x_2\right| < \delta$, therefore,
\begin{eqnarray*}
\left|g_{n} (x_1) - g_{n} (x_2) \right|& =& \left|\int_0^1 \left(\sqrt{x_1+y} -
\sqrt{x_2 + y} \right) f_n(y) \, dy \right|\\
& \leqslant &\varepsilon \int_0^1 \left|f_n(y)\right| \, dy \\
&\leqslant &5 \varepsilon
\end{eqnarray*}
whenever $\left|x_1 - x_2\right| < \delta$. Since the same value of $\delta$ works for all
values of $n$ simultaneously, the family $\{ g_n \}$ is equicontinuous.
Using the uniform bound established above
the conclusion follows from the Arzelà-Ascoli's Theorem.

A: Looks like a Cauchy-Schwarz problem in $L^2$; have you tried that approach?
For part 2, I would try Arzela-Ascoli, since we have boundedness from part 1.
