Incorrect usage of Induction

Induction is often incorrectly used by beginning students by ‘extending it to infinity’ . For example suppose the problem is the following:

Problem: Prove/Disprove that the product of $\mathbb{N}$ number of countable sets is countable.

Often many students show this kind of reasoning: since the product of two sets is countable, and the product of three sets is countable and if we know that the product of $n$ sets is countable then so is the product of $n+1$ sets, therefore we can conclude by induction that the product $\mathbb{N}$ number of countable sets is countable.

The problem with this kind of reasoning is that it is blatantly wrong.

By the same reasoning one can conclude that $\mathbb{N}$ is finite, after all $1$ is finite, $2$ is finite, $3$ is finite and if $n$ is finite then so is $n+1$.

What induction says is that if $P(0)$ holds and if $P(n)\to P(n+1)$ then $P(n)$ holds for all $n\in\mathbb{N}$, it says nothing about whether $P$ holds for $\mathbb{N}$.

One simply can *not* ‘extend to infinity’ like this, understanding that is the first step towards understanding infinity.