Chi-square distribution: proof using induction

[anisotropic]

How would one figure out the following proof using induction?:

The proof can be shown with a chi-squared distribution with n-1 degrees of freedom, but again, a proof using inductive reasoning is what is needed.

 

[anisotropic]

I get the feeling people think I'm trying to get them to "do my homework".

I'm just trying to help a friend who's already put in a lot of work to try and figure this out. I probably would have provided more details, but can't due to not really knowing what this topic is about (I don't understand much of what I entered above).