What is the Limit of Max in a Metric Space?

[cwmiller]

Homework Statement

Prove that $$\rho_{0}(x,y)=max_{1 \leq k \leq n}|x_{k} - y_{k}|=lim_{p\rightarrow\infty}(\sum^{n}_{k=1}|x_{k}-y_{k}|^{p})^{\frac{1}{p}}$$

Homework Equations

The Attempt at a Solution

My approach was to define $a_{m}=max_{1 \leq k \leq n}|x_{k} - y_{k}|$ and $a_{k}=|x_{k} - y_{k}|$. Then since $a_{m} \geq a_{k} \geq 0$ we can replace $lim_{p\rightarrow\infty}(\sum^{n}_{k=1}|x_{k}-y_{k}|^{p})^{\frac{1}{p}}$ with $lim_{p\rightarrow\infty}(\sum^{n}_{k=1}({a_{m} \frac{a_{k}}{a_{m}})^{p}})^{\frac{1}{p}}$.

When trying to break this down I get stuck at $$a_m * lim_{p\rightarrow\infty}(\sum^{n}_{k=1}{(\frac{a_{k}}{a_{m}})^{p}})^{\frac{1}{p}}$$

The limit here should be 1 since $0 \leq \frac{a_{k}}{a_{m}} \leq 1$. However I need to be careful of the case where there are multiple k such that $a_k = a_m$. Does anyone have suggestions for how to proceed? Thanks.

 

[Dick]

The worst case would be ALL of the a_k=a_m, right? Would that change your limit?

 

[cwmiller]

No it wouldn't. Good point. Many thanks.