Question:Every other subspace of V that contains v1,v2,...,vr must contain W?How?

I'm having trouble understanding the following proof.

If are vectors in a vector space , then:

(a) The set of all linear combinations of is a subspace of .

(b) is the smallest subspace of that contains in the sense that every other subspace of that contains must contain .

I understand how to prove (a).

Now I have questions regarding proof (b):

The proof of (b) is given like this:

Each vector is a linear combination of since we can write

Therefore, the subspace contains each of the vectors .

Let be any other subspace that contains

Since is closed under addition and scalar multiplication, it must contain all linear combinations of . Thus contains each vector of

( My Question ) How does the author deduce that contains all linear combinations of in the last sentence of above proof?

How does the statement that " is closed under addition and scalar multiplication" make the statement that " contains all linear combinations of '' true?