Let $\langle M\_i:i<\theta\rangle$ be an increasing chain of Banach spaces, where each $M\_i$ has density character $\mu$ (i.e.,the mininum cardinality of a dense subset of $M\_i$ is $\mu$). Let $B\_i\subset M\_i$ be a dense subset of $M\_i$ of cardinality $B\_i$. Notice that $\bigcup\_{i<\theta}B\_i$ is a dense subset of $\overline{\bigcup\_{i<\theta}M\_i}=:M$, so $density-character(M)\le \mu$. Is it possible to prove that $density-character(M)=\mu$? Thank you.



A question about density character of Banach spaces.
If $X$ is any metric space and $Y$ is any subspace of $X$ then $dc(Y) \leq dc(X)$. 


