You need to show that the binomial expansion of (1+x)^{1/2} is p-adically convergent for |x| < 1. It's sufficient to prove the terms tend to zero: the r-th term is (1/2)(-1/2)(...)(1/2-r) 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.