limits - Sequence of norms

Suppose that we have a sequence of norms $(\|\cdot\|_n)_{n\geq 1}$ on a finite dimensional space $X$ such that, as functions from $X$ to $\mathbb{R}_+$, they point-wise converge to a limit norm $\|\cdot\|_\infty$. Since we are in a finite-dimensional setting, all of the norms are equivalent among them, and, in particular, all of the norms in the sequence are equivalent to $\|\cdot\|_\infty$. An implication of this fact is that there is a sequence of constants $c_n$ such that
$$\forall n \geq 1, \forall x\in X, \quad \|x\|_n \geq c_n\|x\|_\infty.$$

Now, the equivalence constants $c_n$ are not necessarily unique, but, since the norms $\|\cdot\|_n$ approach $\|\cdot\|_\infty$ in the limit, I am wondering how can one prove that they can be chosen to have $1$ as a limit (i.e. $\lim_{n\to\infty} c_n = 1$) ?