Students t-distribution Derivation

[taper100]

I have attempted a derivation of the Students t-distribution probability distribution function in the attached pdf. I defined T to be Z/sqrt(W/v) where Z has standard normal distrubution and W has chi squared distribution with v degrees of freedom. I know that Z and W need to be independent, but I did not use this fact in my derivation. Can someone tell me where I went wrong in the derivation or where I unknowing used this fact in my derivation?

 

[Office_Shredder]

On the first line,
P(T\leq t | W = w) = P(\frac{Z}{c} \leq t )
assumes that Z and W are independent already, since you're saying that the probability that Z is smaller than some number is independent of what our measurement of W is.