[Math] An infinite generating set of a finite dimensional vector space contains a basis

Question

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ?

Please help

Answer 1

You can pick vectors $v_1,v_2,\dots$ out of $S$ such that the elements of $\{v_1,\dots v_k\}$ are independent for $k=1,2,\dots$. That comes to an end because the space has finite dimension. If it can be done for $k=n$ and not for $k=n+1$ then $v_1,\dots v_n$ form a basis, because $S$ is generating.