I came across this proposition in the compressive sensing literature, but a proof wasn't given and I'm not sure how to do it.
Let $\displaystyle x \in R^n$, and let $\displaystyle x_s$ be the best s-term approximation to $\displaystyle x$. Let $\displaystyle 0 < p < q \leq 2$. Then

$\displaystyle ||x-x_s||_q \leq s^{\frac{1}{q}-\frac{1}{p}}||x||_p$.

I can prove it easily in the case where $\displaystyle ||x-x_s||_q \leq ||x_s||_q$ or $\displaystyle n-s < s$, but I can't figure out how to prove it in general. Can anyone help me understand this?