Hi There I have a proof question that I cannot solve Please do help

Let S be the set of functions of the form U= a_{1}sinx +…+a_{n}sin(nx), where a_{1},…,a_{n}are real numbers, a U = (aa_{1})sinx + …. + (aa_{n})sinnx for a is real number

and for V = b_{1}sinx + … +b_{n}sinnx,

U+V = (a_{1}+b_{1})sinx+…+(a_{n}+b_{n})sinnx

Show that S is a vector space