i've been studying the construction of the natural numbers, and i can't solved my own question, my question is:

Why is necessary use the mathematical induction? Let me clarify this, for example, we know if for all $n,m \in \mathbb{N}$ then $n \cdot m = m \cdot n.$ For prove this, we use mathematical induction, but when think about $n,m \in \mathbb{R}$ real numbers, for prove $n \cdot m= m \cdot n,$ don't need mathemathical induction  .

Why sometimes in a proof it's enough take $x \in \mathbb{R}$ arbitrary number but for natural numbers is necessary mathematical induction.

Anyone can help me, please?

I'll hope one give me a hint for understand this question.

I've been studying the construction of the natural numbers, and I can't solve my own question, namely

Why is it necessary to use mathematical induction?

Let me clarify this. For example, we know that, for all $n,m \in \mathbb{N}$, then $n \cdot m = m \cdot n$. In order to prove this, we use mathematical induction, but when we think about $n,m \in \mathbb{R}$ (real numbers), in order to prove $n \cdot m= m \cdot n$, we don't need mathematical induction.

Why sometimes in a proof it's enough to take $x \in \mathbb{R}$, arbitrary number, but for natural numbers mathematical induction is necessary?

Anyone can help me, please?

I hope somebody can give me a hint in order to understand this question.

i've been studying the construction of the natural numbers, and i can't solved my own question, my question is:

Why is neccesary use the mathematical induction? Let me clarify this, for example, we know if for all $n,m \in \mathbb{N}$ then $n \cdot m = m \cdot n.$ For prove this, we use mathematical induction, but when think about $n,m \in \mathbb{R}$ real numbers, for prove $n \cdot m= m \cdot n,$ don't need mathemathical induction .

Why sometimes in a proof it's enough take $x \in \mathbb{R}$ arbitrary number but for natural numbers is necessary mathematical induction.

Anyone can help me, please?

I'll hope one give me a hint for understand this question.

i've been studying the construction of the natural numbers, and i can't solved my own question, my question is:

Why is necessary use the mathematical induction? Let me clarify this, for example, we know if for all $n,m \in \mathbb{N}$ then $n \cdot m = m \cdot n.$ For prove this, we use mathematical induction, but when think about $n,m \in \mathbb{R}$ real numbers, for prove $n \cdot m= m \cdot n,$ don't need mathemathical induction .

Why sometimes in a proof it's enough take $x \in \mathbb{R}$ arbitrary number but for natural numbers is necessary mathematical induction.

Anyone can help me, please?

I'll hope one give me a hint for understand this question.