If $B$ is a basis of $V$ and $U\subseteq V$ is linearly independent then there exists a $C\subseteq B$ such that$ \tilde{B}:= U\dot\cup C$ is a basis

Question

It is an application of Zorn's Lemma, It would help me a lot if somebody could explain one part of the proof that I did not understand

Why can we describe every element of $B$ as a linear combination of Elements in $\tilde{B}$ due to maximality ?

Answer 1

Suppose the opposite, that is, $\exists b\in B$ such that $b\notin span(\tilde{B})$. Call $B'=U \dot{\cup}(C\cup b)$. Since $b$ is not in the span, $C\cup b \subset B$ is a linearly independent set, so $B'\in \mathcal{M}$, but $B'\neq \tilde{B}$ and $B'\geq \tilde{B}$, contradicting the maximality.