real analysis - Are $l_{p} cap k$ and $l_{p} cap k_{0}$ complete in $||$ $||_{infty}$? Are they complete in $l_{p}$ norms?

Let the space $k$ be all convergent sequences of real numbers. Let the space $k_{0}$ be the space of all sequences which converge to zero with $l_{\infty}$ norm. Are $l_{p} \cap k$ and $l_{p} \cap k_{0}$ complete in $||$ $||_{\infty}$? Are they complete in $l_{p}$ norms?

If $k \subseteq l_{\infty}$, with sequence $\{x_{n}\} \in k$, then $\exists \displaystyle\lim x_{n}$. And if $k_{0} \subseteq l_{\infty}$, with sequence $\{x_{n}\} \in k_{0}$, then $k_{0} = \displaystyle\lim_{n \to \infty}x_{n} = 0$.

The help would be appreciated!