
Padic Woes
I've been working on this problem for my analysis class for a while now, and I don't have any idea where to go. Any help at all would be greatly appreciated. Thanks!
Suppose p is an odd prime. Define the set P = {x in Q_p: x_p < 1}, that is, the padic numbers that have padic absolute value less than 1. Define U_1 = P + 1. Prove that the function f(x) = x^2 from U_1 to U_1 is a bijection.

You need to show that the binomial expansion of (1+x)^{1/2} is padically convergent for x < 1. It's sufficient to prove the terms tend to zero: the rth term is (1/2)(1/2)(...)(1/2r) x^r / r! = (+) (2r)! x^r / 2^r (r!)^2. Since p is odd this is an integer (binomial coefficient) times x^r and so goes to zero.