Vector space and maximal proper subspace

Suppose that V is a vector space over F and $\displaystyle \vec{v} \in V $ is any nonzero vector.

Show that there is a subspace W of V maximal subject to the condition

that $\displaystyle \vec{v} \notin W $ ; show that any such W is a maximal proper subspace of V .

(Do NOT assume that V is ﬁnite-dimensional.)

For the first part I thought that I could say :

1. Prove that we can exchange one of the vector in basis of V by $\displaystyle \vec{v}$.

2. Prove that the basis (B) is still one.

3.Let $\displaystyle W = B \backslash \vec{v} $.

4. $\displaystyle B \backslash \vec{v} $ is a basis and thus a maximal independant subset that doesn't contain $\displaystyle \vec{v} $.

5. W is a subspace of V.

Is it correct?

For the second part I don't have a good idea of what to do.

Any hint would be great. Thanks!