# Finite dimensional space norm equivalence

Suppose $V$ is a finite dimensional vector space. We know that all norms on $V$ are equivalent. Suppose the sequence $\{v_i\}$ converges to $v$ in some norm. Since convergence depends only on the topology, is it correct to say that the sequence converges to $v$ in all norms?

Let $\|\cdot\|\,,\;\|\cdot\|'$ denote equivalent norms on a vector space $V$. Assume $(x_n)_n$ has the limit $x$ w.r.t. the metric induced by $\|\cdot\|$. Due to norm equivalence, we have $c>0$ such that $0 \leq \|x - x_n\|' \leq c \|x - x_n\|$ for $n \in \mathbb{N}$...