Why $L^{r}(X)\cap L^{t}(X)\subset L^{s}(X)$ for $1<r<s<t$?
I am working on this homework problem, and I am totally stuck:
Let $(X,\mu)$ be a measure space, and let $1 \leq r < s < t < \infty$. Prove that there exist constants $\alpha,\beta>0$ so that $$ \|f\|_s \;\leq\; \|f\|_r^\alpha\,\|f\|_t^\beta $$ for any measurable function $f\colon X\to\mathbb{R}$. Use this to show that $$ L^r(X) \cap L^t(X) \,\subset\, L^s(X). $$
I know this is supposed not to be difficult. But I cannot solve it.
Hint: Write $\frac{1}{s} = \frac{\alpha}{r} + \frac{\beta}{t}$ and apply Hölder's inequality.