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.
@user7887: I should have added that you want $\alpha + \beta = 1$, so $0 \lt \alpha, \beta \lt 1$. –  t.b. Apr 8 '11 at 6:02
@Theo Buehler: I want to ask how to apply it, the left hand side only has $f$ available, I am thinking I need to make $f$ into $f^{\frac{1}{r}}$ or something, but that looks very ugly. –  Kerry Apr 8 '11 at 6:05