monotonic decrising p norm

Question

I have the following inequality I need to prove:

$$\|x\|_q \leq \|x\|_p \leq n^{\frac{1}{p}-\frac{1}{q}}\|x\|_q$$

For the right inequality, how can i prove $\|x\|_p$ is monotonically decreasing using calculus tools?

That is, $\forall p < q\ \ \exists \|x\|_q \leq \|x\|_p$ given $1\leq p < q \leq \infty\ |\ p,q\in\mathbb N,x\in\mathbb R^n$ .

Is there an easy trick I'm missing?

For the left inequality I thought of using Hölder's inequality but I am not sure how to apply it.

Thank you for any help.

Answer 1

Write $\|x\|_\infty = \max |x_i|$. Clearly $\|x\|_\infty \le \|x\|_p$ for any $p$.

Then for $p < q$ you have $$ \|x\|_q^q = \sum_{k=1}^n |x_i|^q \le \|x\|_\infty^{q-p} \sum_{k=1}^n |x_i|^p \le \|x\|_\infty^{q-p} \|x||_p^p \le \|x\|_p^q.$$

Thus $\|x\|_q \le \|x\|_p$.