Proving a particular inequality using Hölder I'm reading An Introduction to Inequalities by Beckenbach and the following inequality is left as an exercise for the reader ( Chapter 6, p. 104) :
$$[|x|^n + |y|^n]^{1/n} \geq [|x|^m + |y|^m]^{1/m} $$
for all rational $m \geq n \geq 1$. The book states that it can be proven using Hölder's inequality, but I'm having some difficulty doing so.
My attempt is as follows:
\begin{equation*}
|x| = a \; ; \; |y| = b\\
(a^n + b^n)^{1/n} \geq (a^m + b^m)^{1/m}\\
\end{equation*}
Using Hölder we get
$$(a^n + b^n)^{1/n} \cdot (a^m + b^m)^{1/m} \geq a^{n+m}\, + \, b^{n+m} $$
However I don't quite see what I  should do after this. I also tried:
$$(a+b) \cdot (a^{n} + b^{n} )^{1/n} \geq a^{n+1} + b^{n+1}$$
And since $a+b \leq a^{n+1}+b^{n+1}$
$$(a^{n+1}+b^{n+1})(a^{n} + b^{n} )^{1/n} \geq a^{n+1}+b^{n+1} \implies (a^{n} + b^{n} )^{1/n} \geq 1 $$
Doing the same thing for $(a+b)\cdot(a^{m}+b^{m})^{1/m} \geq a^{m+1} + b^{m+1}$ we obtain
$$(a^m+b^m)^{1/m} \geq 1$$
Adding both inequalities
$$(a^n+b^n)^{1/n} \;+\; (a^m+b^m)^{1/m} \geq 2$$
And once again I'm unsure of how to proceed.
 A: Since $m\geq n \geq 1$ are rational, then there are integers $p,q,r,t\in \mathbb{Z}_+$, such that $m=p/q, n=r/t$. Thus, the inequality can be rewritten as
$$
\left(a^{p/q}+b^{p/q}\right)^{\frac{1}{pt}} \leq \left(a^{r/t}+b^{r/t}\right)^{\frac{1}{qr}},
$$
and by making the variable substitutions $a \mapsto a^{qt}, b\mapsto b^{qt},$ then the inequality becomes
$$
\left(a^{pt}+b^{pt}\right)^{\frac{1}{pt}} \leq \left(a^{qr}+b^{qr}\right)^{\frac{1}{qr}},
$$
so we can assume w.l.o.g. that $m,n$ are positive integers.
Now, the case $m=n$ is obvious, and the case $m>n=1$ is also easy to prove by expanding $(a+b)^m$.
So, assume as inductive hypothesis that the inequality holds for $m-1 > n-1 >1$ for all $a,b$ nonnegative, then using Hölder's inquality and the inductive hypothesis:
$$a^m + b^m = aa^{m-1}+bb^{m-1}\leq \left(a^n + b^n\right)^{1/n}\left(a^{(m-1)\frac{n}{n-1}} + b^{(m-1)\frac{n}{n-1}}\right)^{\frac{n-1}{n}} \leq \left(a^n + b^n\right)^{1/n}\left(a^{(n-1)\frac{n}{n-1}}+b^{(n-1)\frac{n}{n-1}}\right)^{\frac{(m-1)(n-1)}{n(n-1)}} = \left(a^n + b^n\right)^{m/n},$$
thus, the inequality is proven by induction.
The same technique can be used for any number of variables, and the inequality can be extended to real $m,n$, by taking rational sequences that converge, and using the continuity of $f(t) = \left(\sum_{k=1}^n |x_k|^t\right)^{1/t}$.
