Baby Analysis Question
I'm reading through Rudin and got to this proof, which I can't understand: Prove that if .
The proof goes: Assume , then . Let , then by the binomial theorem . It's this last inequality that I don't get. If what we're doing is dividing each th term by to get , n-many times, then how do we know that we're decreasing the expression, since it's possible that ?
Well the binomial theorem gives you this:
In our case, we put and , giving:
Now the RHS of this equation has more than two terms, and all terms are positive (because , so just by its definition). Drop all the terms of the series except when k=0 and k=1. The RHS will be at least as large as this because it's equal to it plus all the terms we dropped. This will give you that: