I am kindly asking for someone to give me some feedback on my proof:
The proposition is: "Prove that V is infinite-dimensional iff there exists a sequence $v_1,v_2,...$ of vectors in V such that $v_1,v_2,...,v_m$ is linearly independent for every positive integer $m$".
I already did and checked $(\Rightarrow)$.
The other way is the one i am asking for feedback. My proof is by contradiction:
Suppose V is finite-dimensional. Then there is a list of vectors $v_1,v_2,...,v_k$, for some positive integer k such that $Span(v_1,v_2,...,v_k)=V$. Therefore there exist $a_1,...,a_k$ in F such that $a_1v_1+a_2v_2+...+a_3v_m = v_{k+1}$, for some vector $v_{k+1}$ in V. But then $v_1,v_2,...,v_k,v_{k+1}$ will not be linearly independent. We arrived to a contradiction. Therefore V is infinite-dimensional.
I am aware of similar questions such as: An exercise about the proof of an infinite-dimensional vector space , Show that a infinite dimensional vector space must have linearly independent vectors for every positive integer
etc.
But if found none including my prove.
Thank you.