Is convergence of of vector in $\mathbb{R}^k$ equivalent to the elementwise convergence even in nonstandard distance?

Question

I'm wondering, for $x_n \in \mathbb{R}^k$, whether $x_n \to x$ iff $x_n^p \to x^p$, for $p=1,\cdots,k$.

The equivalence is direct for the standard Euclidean norm, norm defined by maximum of it component, and $L2$ norm.

If a distance satisfies the symmetry and triangle inequality, does this distance also means the two conditions are equivalent?