Let $X$ be a normed linear space with two norms $||\cdot||_1$ and $||\cdot||_2$. Prove or disprove that this statements are equivalent:
- $||\cdot||_1$ and $||\cdot||_2$ are equivalent,
- $\{x_n\}$ converges in $||\cdot||_1$ iff $\{x_n\}$ converges in $||\cdot||_2\;\; $ $\forall \{x_n\}\in X$
The first implies the second trivially because of equivalence of topologies. But it implies something more: sequence converges to the same element in both norms. But I don't have such condition.
Is this a necessary condition to converge to the same in both norms, or this is just a consequent of "if and only if" in the statement of convergence, or there is a example disproving the equivalence?