[Math] extending a linearly independent set to a basis

Question

I want to show that every linearly independent set in a finite-dimensional linear space can be extended to a basis for the entire space.

Answer 1

Every linearly independant set has at most $n$ elements in a vector space $E$ s.t. $\dim E=n$, if this set does not spans $E$ then there's $x\in E$ which isn't a linear combination of elements of this set so subjoin this element $x$ to the set and it remains linearly independant, we can repete this procedure until we have $n$ elements