# Minkowski inequality proof without using Holder's inequality

We are told to proof the Minkowski inequality specialized to sequences:

$$\left(\sum_{i=1}^n|a_i+b_i|^p\right)^{1/p}\leq \left(\sum_{i=1}^n|a_i|^p\right)^{1/p} +\left(\sum_{i=1}^n|b_i|^p\right)^{1/p}$$

when $$p\geq1$$

Well, the problem is that all the proofs I have found "use" the Holder's inequality to proof that, but we haven't seen it in class yet so I can't "use" it to proof the Minkowski inequality.

The proof must somehow "elementary", with things we have already learnt in class. "Cauchy-Schwartz inequality" and the "Triangle inequality"are two of the inequalities we have learnt, for example.

Could someone help me? (and the rest of the students in my class, who are in the same situation).

For all $$1\leq i\leq p,$$ |$$a_i+b_i$$|$$\leq$$|$$a_i$$|+|$$b_i$$|. Then we have that |$$a_i+b_i$$|$$^p$$ $$\leq$$ (|$$a_i$$|+|$$b_i$$|$$)^p.$$ Hence $$\sum_{i=1}^{p} |a_i+b_i|^p\leq\sum_{i=1}^{p}(|a_i|+|b_i|)^p$$. Therefore $$(\sum_{i=1}^{p} |a_i+b_i|^{p})^{1/p}\leq(\sum_{i=1}^{p}|a_i|^p+\sum_{i=1}^{p}|b_i|^p)^{1/p}\leq(\sum_{i=1}^{p}|a_i|^p)^{1/p}+(\sum_{i=1}^{p}|b_i|^p)^{1/p}.\blacksquare$$
• Why do you first say $1\leq i\leq p,$? Then you proof with $\sum_{i=1}^{p}$ and not $\sum_{i=1}^{n}$ and i am not sure if something changes. And could you please explain better the last step? – Andarrkor Oct 18 '18 at 15:32
• $(\sum_{i=1}^{p} |a_i+b_i|^{p})^{1/p}\leq(\sum_{i=1}^{p}|a_i|^p+\sum_{i=1}^{p}|b_i|^p)^{1/p}$ is this step correct? – Andarrkor Oct 22 '18 at 15:16