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!
Observe that for $p<\infty$ $$ l_p\cap k = l_p \cap k_0 = l_p. $$ Hence these spaces are complete with respect to the $l_p$ norm.
To see these inclusions: If $x\in l^p$, then $\sum_{n=1}^\infty |x_n|^p$ converges, hence $\lim_{n\to\infty}x_n=0$ and $x\in k_0$.
These spaces are not complete with respect to the $l_\infty$ norm: Take the sequence $x_n$ defined as $$ x_n = (1^{1/p}, 2^{1/p}, \dots , n^{1/p}, 0, \dots) \in k_0. $$ This is a Cauchy sequence in $l^\infty$, but the sequence is unbounded in $l^p$, hence cannot converge.
If $p=\infty$, then $k_0\subset k \subset l_\infty$. Moreover, $k_0$ and $k$ are closed subspaces of $l_\infty$.