Prove 1 is the smallest natural number via induction.

Question

Basically I am stuck at the step where I need to prove that n + 1 >= 1, where n is a natural number. From the assumption, we know that n >= 1. Not sure how to continue from there.

Answer 1

The claim you're trying to prove is that for all $n \in \mathbb{N}$, $n \ge 1$.

It seems like you've got the base case, which follows pretty quickly from reflexivity.

For the inductive case, assume $k \ge 1$. We know that $1 \ge 0$ (hopefully this is a fact you can use), and so $k + 1 \ge k$. Transitivity tells us that $k + 1 \ge 1$.