By definition-

If V be a vector space over an arbitray field F, then we say that V is finite dimensional if it is spanmed by a finite set of vectors.

Let, dimV =n

\(\displaystyle\Rightarrow\) V is spamed by a set of n linearly independent vectors in V,

say \(S=\{v_1,v_1,v_1,...,v_n\}\)

Now, as W is a stubspace of then W is spaned by at most n elements of the set S.

Hence, by definition of finite dimensional vector sapace- W is finite dimensional.