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.