In calculus class, I saw a proof of this, but I am not convinced.
Note:
A$\subseteq \Bbb{R}$ is inductive if and only if 1$\in A$ and $\forall x \in A \Rightarrow x+1 \in A$.
I tried to do a proof here:
Suppose that $A$ is a inductive set.
By definition, $\Bbb{N}$ is the untersection of all inductive sets.
That is: $$\Bbb{N} = \bigcap A_i\in I$$ where $A_i$ is a inductive set.
This is the same as: $$\Bbb{N} = \{x: x\in A_1 \land x\in A_2 \land ... \land x\in A \land ... \}$$
Thus, $x \in \Bbb{N} \Rightarrow$ x $\in A$.
By definition of a subset, $\Bbb{N} \subseteq A$.
Is my proof correct?
