Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is. Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff  $(V,||\cdot||_2)$ is. I really didn't get it. Of course both spaces are normed spaces but there are two things I don't understand:


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*Why is it so important to look at "Cauchy sequences"? Why can't I look at converging sequences? Aren't those equivalent?

*If those are equivalent, ones a sequence converges to a limit (which is in the space) should it be different in the other "space", that is $(V,||\cdot||_2)?
I really didn't understand what I need to do here. I would truly appreciate your help.

It suffices to show that 


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*a sequence $\left\{v_n\right\}$ is Cauchy in $\left\Vert\cdot\right\Vert_1$ iff it is Cauchy in $\left\Vert\cdot\right\Vert_2$. 

*a sequence $\left\{v_n\right\}$ converges in $\left\Vert\cdot\right\Vert_1$ iff it converges in $\left\Vert\cdot\right\Vert_2$. 
Why is this sufficient?  Because then the argument goes like this: Suppose $\left\Vert\cdot\right\Vert_1$ is complete; that is, $V$ is a Banach space in the $\left\Vert\cdot\right\Vert_1$ metric.  Let $\left\{v_n\right\}$ be a Cauchy sequence in $\left\Vert\cdot\right\Vert_2$.  By (1.) the sequence is Cauchy in $\left\Vert\cdot\right\Vert_1$.  Since $\left\Vert\cdot\right\Vert_1$ is complete, the sequence converges in $\left\Vert\cdot\right\Vert_1$, and by (2.) the sequence converges in $\left\Vert\cdot\right\Vert_2$.  Therefore $\left\Vert\cdot\right\Vert_2$ is complete.
