I'm reading through Rudin and got to this proof, which I can't understand: Prove that $\displaystyle \displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{p} = 1$ if $\displaystyle p > 0$.

The proof goes: Assume $\displaystyle p > 1$, then $\displaystyle \sqrt[n]{p} - 1 > 0$. Let $\displaystyle x_{n} = \sqrt[n]{p} - 1$, then by the binomial theorem $\displaystyle (x_{n} +1)^{n} = p \geq 1 + nx_{n}$. It's this last inequality that I don't get. If what we're doing is dividing each $\displaystyle x_{n}^{n-i}$th term by $\displaystyle x_{n}^{n-i-1}$ to get $\displaystyle x_{n}$, n-many times, then how do we know that we're decreasing the expression, since it's possible that $\displaystyle x_{n} < 1$?