Show that $\lim_{n \to \infty} \int_{0}^{1}|f_n(x)| \, dx= 0$ if $\int_0^1|f_n(x)|^2\,dx < 100$

Question

Let $\{f_n\}$ be a sequence of Lebesgue integrable functions such that

(i) $\int_0^1|f_n(x)|^2 dx < 100$

(ii) $ f_n \to 0$ almost everywhere

We must show that

$$ \lim_{n\to\infty}\int_0^1 |f_n(x)| \, dx = 0$$

I have a solution using Egoroff and Schartz Inequality. Is that necessary? Any other ideas ? Also I prove it by myself without that. I will edit the post later.

Answer 1

Hint: By the Cauchy-Schwarz inequality,

$$\begin{align*} \int_0^1 |f_n(x)| \, dx &= \int_0^1 1_{\{|f_n(x)| \leq \epsilon\}} |f_n(x)| \, dx + \int_0^1 1_{\{|f_n(x)|>\epsilon\}} |f_n(x)| \, dx \\ &\leq \epsilon + \sqrt{\int_0^1 1_{\{|f_n(x)|>\epsilon\}} \, dx} \cdot \underbrace{\sqrt{\int_0^1 f_n(x)^2 \, dx}}_{\leq 10}. \end{align*}$$

Now let first $n \to \infty$ and then $\epsilon \to 0$.