I want to show the following:
On a set of real numbers that has finite measure, if $a > b$ and $||f_n - f||_a \rightarrow 0$, then $||f_n - f||_b \rightarrow 0$ as well.
Perhaps I should be using Holder to say that $||f_n - f||_a \geq ||f_n - f||_b$, but I'm not sure how to do so.
$\int |f_n-f|^{b} \leq (\int |f_n-f|^{b})^{a/b})^{b/a} (\int 1^{t} )^{1/t}$ where $t=\frac a {a-b}$. (Note that $\frac 1 t +\frac b a =1)$.