Attempting to prove closure under multiplication. Is this correct?

Proof of closure under scalar multiplication

Multiply vector x by scalar k the result is a real value that is in subspace W. 3 * x = 3x

K* (x , 3x) = (kx , k*3x) by multiplication of a vector and a scalar (kx , k*3x) = (kx, 3*kx) By the commutative property of multiplication

The coordinate (kx, 3*kx) is in W since the components,

the scalar K and the vector x, 3x are all real when it is multiplied by a real number the result is in subspace W.