how do i prove that every subspace is a spanning set for itself?

sorry im very weak at proofing.

my working:

let S be the subspace.

so we need to show that S = span(S)

we shall show that S⊂span S ( shown in previous post) and span S ⊂S

to show that spanS ⊂ S,

lets assume that S is in n dimension where xi∈ S where 1<i<n

span S =a1x1+...+anxn ⊂ S as S being a subspace obeys the laws of scalar multiplication and addition.

is this done like this?