Prove that for a positive number only exists one natural number such that $b{++} = a$

Question

My first time on Mathematics, thank you for your attention and patience.

Eventually, I decided to have a taste of mathematical analysis, useful for my University. My weapon of choice is Terence Tao's Analysis 1. I love its style. It is making me clear a lot of things that before were cause of headaches. Unfortunately, it does not provide exercise solutions allowing me to test the product of my efforts.

I am trying to solve the Exercise 2.2.2:

Let $a$ be a positive number. Then there exists exactly one natural number $b$ such that $b{++}=a$.

The author suggests to use induction. Frankly, I have no idea how to induct on such a problem. My approach used another way:

1. $a \neq 0$ by definition of positive natural number;

2. Peano's Axiom 4 states that different natural numbers must have different successors;

3. Hence, if $b{++} = a$ and $c{++} = a$ then $b = c$, contradicting the lemma stating that $\exists! (b{++} = a)$.

   The same problem is faced in this question, but I found it quite confusing. Is my approach acceptable? How should induction work on such a problem?

Answer 1

You have uniqueness as a consequence of axiom 4.

In order to show existence, I'd still suggest to use induction (though it is a very trivial induction proof that does not even use the induction hypothesis)