Prove that if $W$ is a subspace of a vector space $V$ and $w_1, w_2, ..., w_n$ are in $W$, then $a_1w_1 + a_2w_2 + ... + a_nw_n \in W$ for any scalars $a_1, a_2, ..., a_n$.
My solution is we have $a_iw_i \in W$ for all $i$. And we can get the conclusion that $a_1w_1, a_1w_1 + a_2w_2, a_1w_1 + a_2w_2 + a_3w_3$ are in $W$ inductively.
Any ideas on how to improve this because I feel it is not enough.
Thank you in advance.