As the title says. I encounter this problem in Bernd Schroeder's book of "Mathematical Analysis: A Concise Introduction", p.15. It essentially characterizes natural number from the axioms regarding real number, i.e. the axioms of addition and multiplication of $\mathbb{R}$ as a field. My attempt is the following using Principle of Induction.
Let $S:=\{n \in \mathbb{N}: n \ge 1\}$. Check if $ 1 \in S$. Since $1 \ge 1$ holds, therefore $1 \in S$. Suppose $n \in S$, we want to show $n+1 \in S$. $n+1 \ge 1 \Rightarrow n \ge 0$.However from the induction step, we have $ n \ge 1 > 0$, therefore $n+1 \in S$. By Principle of Induction, $S=\mathbb{N}$ $\Box$
EDIT1: There is a mistake in the proof as pointed out by talmid.
EDIT2: The author of the above mentioned book constructs natural number, integers and rationals from Completeness Axiom. A similar approach can be found in Royden's Real Analysis