This problem has two parts:
(a) If $f\in L^1[0,1]\cap L^2[0,1]$, then $\|f\|_1 \le \|f\|_2$.
(b) Use (a) to deduce that $L^2[0,1]$ is a subset of $L^1[0,1]$.
Without using part (a), let $f$$\in$$L^2[0,1]$. Since the constant function $1$ $\in$ $L^2[0,1]$, by Holder inequality, we can conclude that $\|f\|_1$$\le$$\|f\|_2\|1\|_2$$=\|f\|_2$.
But how can you deduce part (b) from part (a)? Also, how do you prove the claim in part (a)?