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 x \in R^n, and let x_s be the best s-term approximation to x. Let 0 < p < q \leq 2. Then

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

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