Linearly Independent Set is Contained in some Basis/Finite Dimensional Case/Proof 1

Proof
By hypothesis there is a basis $B$ of $E$ with $n$ elements.

Then $H \cup B$ is a generator for $E$.

So by Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set there exists a basis $C$ of $E$ such that $H \subseteq C \subseteq H \cup B$.