If I have a statement $P(n)$, and I show that $P(1)$ is true and $P(k)$ being true implies $P(k+1)$ is true, then it is my understanding that induction shows that $P(n)$ is true for all natural numbers.
Suppose I have the statement $P(n)$ being "$\sum_{i=1}^{n}i$ is finite"
Clearly for $n=1$ the sum is finite, and if $n=k$ is finite that $n=k+1$ is also finite, so I can conclude that the sum is finite for all natural numbers.
If I extend my index to include "infinity" and I define $P(\infty)$ to be $\sum_{i=1}^{\infty}i$, it is not finite (I think?).
It seems that induction only works for finite numbers, so I can't use it to make conclusions for even a countably infinite number.
Is this true?
I was going to use induction for a problem where I needed to show a statement about $\bigcup_{i}^{}$ was true for a countably infinite index, but it seems like induction is not usable.
As you indicated, the proof technique of mathematical induction is used
to prove that a property holds for every natural number $n$.
It does not prove that the property holds when $n$ is replaced by $\infty.$
Your counterexample "$\sum\limits_{i=1}^n i$ is finite" illustrates that.