TITLE: Is there a null sequence that is in not in $\ell_p$ for any $p<\infty$?
QUESTION [5 upvotes]: Is 
$$\bigcup_{p<\infty}\ell_p=c_0 ?$$
At least one inclusion obvious: every $p$-summable sequence converges to zero.

REPLY [4 votes]: Prove: If $V$ is a separable Banach space, then $V$ cannot be written as a countable union of proper linear subspaces.  
Because of inclusions among the $\ell_p$ spaces, your union is, in fact, equal to a countable union.