I need to show a few things and I know a bit of how to do it but I can't write it out properly

I have to give a lecture about it in 3 hours for my honors seminar course so I need to have this pretty much perfect

here goes

step 1

show that every finite dimensional pre-Hilbert space (inner product space) V with dim V = k is isomorphic to $\displaystyle \mathbb{C}^k$

my prof told me something about orthogonal bases using the gram-schmidt process but I was never very good at linear algebra

step 2

using this and the knowledge that every cauchy sequence in $\displaystyle \mathbb{C}^k$ is convergent show that every cauchy sequence in V is convergent

this shows that V is complete and thus is a Hilbert space

so I think I have the reasoning behind it all worked out but none of the actual math part of the proof

can someone help me?

I have a backup proof of the completeness of finite dimensional pre-hilbert spaces using induction on the dimension but it doesn't cover part 1 and doesn't give us as powerful of results

edit:

lecture went fine, class was 3:30 - 4:50 and at 4:45 I started stating the theorem, then the professor cut me off and proceeded to just quickly outline the proof

so I didn't have to haha

I still have to finish next tuesday but I don't have to go over this one :P