Equivalence of Metrics in R^{n}

[muzak]

Homework Statement

Prove that in R$^{n}$, the euclidean metric, the d$_{\infty}$=max{|a1-b1|,...,|a$_{n}$}-b$_{n}$|}, and d = |a1-b1|+...|a$_{n}$}-b$_{n}$|.

Homework Equations

Uniform Equivalence: basically p,d so that we have the two inequalities with some constants like p(x,y)$\leq$Ad(x,y) and d(x.y)$\leq$Bp(x,y).
Schwarz inequality.

The Attempt at a Solution

I was going to do this in straightforward manner but when we go to see what our constants are, they turn out to n or $\infty$. I don't know what to do. Can we treat them as coefficients in the two respective ineqaulities?

 

[micromass]

Well, n is a good constant, but $\infty$ is not. Where did you get $\infty$?? Then we'll look if we can fix that.