Condition number of simple eigenvalues

Hi, there's a step in the derivation of the condition number that I do not understand.

Let and let be a simple eigenvalue of .

Let be a right eigenvector and perturbe such that

I multiply this thing and use the fact that to get,

Now let be a left eigenvector of and multiply the above equation from the left. Using the fact that we can write

hence,

By assuming that we can bound this as

I hope you are still awake :) What happens next is confusing me. If then

*and therefore*

I do not understand the (*and therefore*) part :) Is this some alternative definition of condition number? Hope someone has the patience to help me out, thanks!