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)\leqAd(x,y) and d(x.y)\leqBp(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.