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 $densitycharacter(M)\le \mu$. Is it possible to prove that $densitycharacter(M)=\mu$? Thank you.
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If $X$ is any metric space and $Y$ is any subspace of $X$ then $dc(Y) \leq dc(X)$. 

