Mainly I just want verification if this is correct or otherwise.

Consider two subspaces V and W of R_n.

Is V U W necessarily a subspace of R? Is the intersect of V and W a subspace?

Yes to both?

If a vector x is an element of V intersect W, then it is an element of V, then therefore all linear combinations are also in V. It is also in W, and therefore all linear combinations are in W as well. Thus all linear combinations are in V intersect W. The other one would use similar logic.

ALSO

Consider vectors v1, v2... v_m in |Rn. Is the span of the vectors necessarily a subspace of Rn?

I think this is true but how would I justify it?

Say that span is all linear combinations, and then c can be 0 so that fulfills one requirement.

Then say that all scalar multiples of any vectors with real numbers will result in vectors of real numbers?