Induction for infinite union of countable sets

[idle juniper]

Task b
Can someone explain why induction cant be used to prove this
I have read that induction proves this for any given finite n but why doesnt this extend to the infinite case

[misty hemlock]

induction to prove a statement P(n) is only useful for finite n. there are plenty of cases where P(n) can be proved inductively for all natural numbers, but P(infinity) is false. for example the statement P(n), "n is finite", is true for all natural numbers n, but not true for P(infinity)

[idle juniper]

doesnt induction cover all natural numbers though and since the natural numbers are infinite shouldnt it work

[misty hemlock]

i see exactly what you mean, the first question to think about would be 'is infinity a natural number?'

you're right, there are infinitely many natural numbers, but infinity is not itself a natural number

so by proving some statement for all the natural numbers, you have proven it to be true for infinitely many numbers, but not for infinity

does that make sense?

[idle juniper]

I think so

Thank you

[misty hemlock]

you're very welcome, is there anything else i can do to help?

[idle juniper]

No that was it 🙂

.solved