What is (your) definition of a vector subspace? Mine is that W is a subspace if W is a subset of V and for all x and y in W and s and t in R (or whatever the underlying field is, perhaps C) the sx+ty is in W and 0 is in W. (note this is redundant by settinf s=t=0)
What is the span of a subset? iti s the set of all combinations
for t_i in R (or the underlying field) and x_i in W
so W=span(W) means exactly that all finite combinations of elements of W are in W.
so we are trying to show that
W closed under combining (adding up) two elements of itself if and only if W is closed under combining a finite number of elements of itself.
Obvioulsy one way is simple: if i can add up any number of combinations of elements then i can in particular add up two of them. Conversely...?