Proof of span(S) is a subspace:a question

We know the definition of subspace which is:

If is a set of one or more vectors from a vector space , then is a subsace of if and only if the following conditions hold:

(a) If and are vectors in , then is in .

(b) If is any scalar and is any vector in , then is in .

Also the definition of is:

If is a set of vectors in a vector space , then the subspace of consisting of all linear combinations of the vectors in is called the space spanned by , and we say that the vectors span . To indicate that is the space spanned by the vectors in the set , we write:

or

Now the problem I'm having is how is it possible that is a subspace? I can't make a connection with the definition and the proof.

Is it possible to explain the proof of why is a subspace?