Gram-Schmidt process vectors

Prove that if $\displaystyle \{ w_{1}, w_{2}, ... , w_{n} \} $ is an orthogonal set of nonzero vectors, then the vectors $\displaystyle v_{1}, v_{2}, . . . , v_{n} $ derived from the Gram-Schmidt process satisfy $\displaystyle v_{i} = w_{i} \ \ \ \forall i $

my proof so far:

I intend to use induction.

Now, $\displaystyle v_{1} = w_{1} $ is travial.

Suppose that n = k is true, then I have $\displaystyle v_{k} = w{k-1} - \sum ^{k-2}_{j=1} \frac {<w_{k-1},v_{j}>}{ || v_{j} || ^2 } v_{j} = w_{k} $

Now, how would I use that information as well as the fact that the vecters are orthogonal to get k+1 is true?

thanks