I'm trying to find the error in this induction proof:
We want to prove that for all elements $a$ such that $a \in \mathbb{Z}^+$, that $a^2 \leq a$.
The base case is when $a = 1$, therefore $1^2 = 1$ and we know this is true.
The induction case:
Let $b \in \mathbb{Z}^+$. The induction hypothesis states that $b^2 \leq b$. Therefore, we have to show that $(b + 1)^2 \leq b + 1$.
$b^2 \leq b^2 + 2b = (b^2 + 2b + 1) - 1 = (b + 1)^2 - 1$
Since $(b + 1)^2 \leq b + 1$, we know that
$(b+1)^2 - 1 \leq (b + 1) - 1 = b$
Thus we conclude $b^2 \leq b$ by the induction hypothesis.
My thinking:
I know the proof forgot to specify that $b \geq 1$ in the induction step. I'm struggling to see how this matters downstream.