Applying equivalence of norms on $\mathbb R^n$ . Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequance on $\mathbb R^n$ converges  to an element $x \in \mathbb R^n$ under the $\|\cdot\|_2$ norm if and only if the sequance converges to $x$ under the $\|\cdot\|$ norm.
I want to say that $c\|\cdot\| \leq \|\cdot\|_2 \leq d\|\cdot\|$ given $c,d \in \mathbb R$ since $\mathbb R^n$ is a closed space and than to show c=d. is that a correct way? how can i show d=c?

First of all, $c$ is not equal to $d$.
If $x_n\to x$ under $||\cdot||_1$ then for all $\epsilon>0$ there is an $N$ such that for all  $n>N$, we have $$||x_n-x||_1<\epsilon.$$
Now from
$$c\|x_n-x\|_2 \leq \|x_n-x ||_1 \leq d\|x_n-x\|_2,$$ it is clear that 
$$\|x_n-x\|_2 < \frac{\epsilon}{c}.$$  One can deduce that $x_n\to x$ under $\|\cdot\|_2$ and reciprocally. 
