Simple proof of Complex Inner Product Space

[RJLiberator]

Homework Statement

Prove that <v|0>=0 for all |v> ∈ V.

Homework Equations

The Attempt at a Solution

This is a general inner product space.

I break it up into 2 cases.
Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0.

Case 2: If |v> =/= 0 then:
I use <v|0> = Σv_i * 0_i
and from here it is clear to see that the sum adds up to 0 as every component is multiplied by the 0 vector.

My question: Is this a safe definition of the complex inner product? Am I OK to use the summation definition in this general proof?
Second Question: Is the proof correct? Any reason why the 0 vector would need to be proven further to sum the components to 0?

Thanks

 

[RJLiberator]

I think I have a better idea.

<v|0> = <v|0v> (by multiplication by 0.)
= 0<v|v> by inner product axiom 2
=0
and done!

This is the scalar 0, and not the vector 0 as stated in the question.