Cauchy Sequences Lemma in Vector Space E

Question

I ran into a Lemma.

Suppose $||.||_1$ and $||.||_2$ are two norms in vector spapce E, such that $||.||_1$ and $||.||_2$ are equivalent norms and {$x_n$} is an equivalent in E, then {$x_n$} is cauchy sequence in (E, $||.||_1$) if and only if {$x_n$} is cauchy sequence in (E, $||.||_2$).

my friends would help me, how prove this lemma? i try to familiar with cuachy sequences recently.

Answer 1

If $||.||_1 \sim ||.||_2$ (are equivalent) then there exists two constants $c, C > 0$ such that for all $ x \in E$ we have:

$$c||x||_1 \leq ||x||_2 \leq C||x||_1$$

Now if $(x_n)$ is cauchy with respect to $||.||_1$, it means that given $\varepsilon > 0$, we have $$||x_n - x_m||_1 < \dfrac{\varepsilon}{C}$$ if $n,m$ are bigh enough. But this means that

$$||x_n - x_m||_2 \leq C||x_n - x_m||_1 < \varepsilon$$