Prove that a sequence in a set is Cauchy in one metric if it is Cauchy under an equivalent metric

Question

Let d and d' be strongly equivalent metrics on a set X. Prove that a sequence is Cauchy in (X,d) if and only if it is Cauchy in (X,d').

d is strongly equivalent to d' if $\exists$ constants $c_1,c_2$ such that for any $p,q$ we have: $d(p,q)\leq c_1 d'(p,q)$ and $d'(p,q)\leq c_2 d(p,q)$

Please add definition of "strongly equivalent". – coffeemath Feb 19 '18 at 11:54 — coffeemath, Feb 19 '18 at 11:54

score 1 · Accepted Answer · answered Feb 19 '18 at 12:12

1

Suppose $(x_n)$ is Cauchy with respect to $d$. Fix $\varepsilon > 0$. Then there exists an $N$ such that $$n, m \ge N \implies d(x_n, x_m) < \frac{\varepsilon}{c_2} \ldots$$

answered Feb 19 '18 at 12:12

Theo Bendit

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Prove that a sequence in a set is Cauchy in one metric if it is Cauchy under an equivalent metric

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