Quote by Dick
V is infinite dimensional. Pick a vector v1 in V. If v1 spanned V, V would be finite dimensional. It's not. So there's another independent vector v2. If {v1,v2} spanned then V would be finite dimensional. It's not. So there's an independent v3. If {v1,v2,v3} spanned then etc. It's sort of induction. The point is that any set of linearly independent vectors can always be extended.

Thank you very much, I get it.
How about the backward direction? Is it correct? Does it need tweaking? Or is it completely wrong?