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Minkowski's inequality says the following: For every sequence of scalars $a = (a_i)$ and $b = (b_i)$, and for $1 \leq p \leq \infty$ we have: $||a+b||_{p} \leq ||a||_{p}+ ||b||_{p}$. Note that $||x||_{p} = \left(\smash{\sum\limits_{i=1}^{\infty}} |x_i|^{p}\right)^{1/p}$. This is how I tried proving it: \begin{align*} ||a+b||^{p} &= \sum |a_k+b_k|^{p}\\\ &\leq \sum(|a_k|+|b_k|)^{p}\\\ &= \sum(|a_k|+|b_k|)^{p-1}|a_k|+ \sum(|a_k|+|b_k|)^{p-1}|b_k|. \end{align*}

From here, how would you proceed? I know that you need to use Hölder's inequality. So maybe we can bound both the sums on the RHS since they are products.

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    instead of using `$\displaymath...$` you can use `$...$` to get displayed math; and macros like `\begin{align}` to get aligned equations.2011-01-05

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Holder's Inequality would say that $\sum |x_ky_k| \leq \left(\sum |x_k|^r\right)^{1/r}\left(\sum|y_k|^s\right)^{1/s}$ where $\frac{1}{r}+\frac{1}{s}=1$.

Apply Holder's twice, once to each sum, using $x_k = a_k$, $y_k = (|a_k|+|b_k|)^{p-1}$ in one, and similarly in the other, with $r=p$ and $\frac{1}{s}=1-\frac{1}{p}$.

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    I think we should apply holder to $y_k=(|a_k+b_k|)^{p-1}$ and OP should not apply triangle inequality to $|a_k+b_k|^{p-1}$. Thanks2016-12-24