MHB Real Analysis Help: Metric Spaces

[wonguyen1995]

Show that two metrics p and T on the same set X are equivalent if and only if there is a c > 0
such that for all u,v belong to X,
(1/c)T(u,v)=<p(u,v)=<cT(u,v)

Please help me , I'm so confused about Real Analysis.

 

[Opalg]

Show that two metrics p and T on the same set X are equivalent if and only if there is a c > 0
such that for all u,v belong to X,
(1/c)T(u,v)=<p(u,v)=<cT(u,v)

Please help me , I'm so confused about Real Analysis.

What definition of equivalence for metrics are you using? The usual definition is that two metrics are equivalent if convergence of a sequence in one metric implies convergence in the other metric. That does not imply the condition $(1/c)T(u,v) \leqslant p(u,v) \leqslant cT(u,v)$ (for all $u,v\in X$), which is usually called strong equivalence of the two metrics (see the discussion in the Wikipedia page that I linked to above).