Uniform Convergence Implies $L^2$ Convergence and $L^2$ Convergence Implies $L^1$ Convergence

If $f_n$ converges to $f$ uniformly (i.e. $\sup_{x \in I}|f_n(x)-f(x)| \to 0$), then $$\|f_n-f\|^2_2=\int_I |f_n(x)-f(x)|^2\,dx \leq m(I) \left(\sup_{x \in I}|f_n(x)-f(x)|\right)^2 \to 0,$$ so $f_n$ converges to $f$ in $L^2$.

If $f_n$ converges to $f$ in $L^2(I)$, then by Hölder inequality we have $$ \|f_n-f\|_1=\int_I |f_n(x)-f(x)|\,dx \leq (m(I))^{\frac12} \|f_n-f\|_2 \to 0, $$

so that $f_n$ goes to $f$ in $L^1$.

Note that these results (with the same proof) hold in a much more general context, as pointed out in the other answer, I tried to give the most "hands on" proof possible.

On a finite measure space $L^q$ convergence implies $L^p$ convergence for any $1\leq p \leq q \leq \infty$. Take $\infty>q>p$, so $q/p > 1$.This follows from Holder's inequality, $$ ||f||_p^p = \int |f|^p\,d\mu = \int |f|^p \cdot 1\,d\mu \leq \left(\int (|f|^{p})^{q/p}\,d\mu\right)^{p/q}\left(\int 1\,d\mu \right)^{1-p/q}= ||f||_q^p \mu(X)^{1-p/q} $$ so we see that if $f \to 0$ in $L^q$ then $f \to 0$ in $L^p$.

The case $q=\infty$ is easier, just using $|f| \leq ||f||_\infty$ almost everywhere, $$ ||f||_p^p = \int |f|^p\,d\mu \leq \mu(X) ||f||_\infty^p $$ and again the result follows.