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Q:
Is the following «infinite» sequence valid or not?
Consider the following sequence:
$$
f_n = 1 + frac{1}{2 + frac{1}{2 + frac{1}{2 + frac{1}{2 + ldots}}}}
$$
This sequence contains an internal division by zero. Is this sequence, as a set, a countable infinite set or an uncountable infinite set? If it’s the former, I just want to know why and if it’s the latter, what can be said about it and what sets can’t be unioned together.
Edit: I know that
$$
lim_{n to infty} f_n = frac{1}{1+frac{1}{1+frac{1}{1+ldots}}} = -1
$$
However, I was wondering about the sets, and whether they were finite or countable. As explained in other answers, infinite sets that don’t contain an infinite ascending chain (such as the sets in Russell’s paradox) are not countable. However, I was wondering if this particular sequence could be a countable infinite set, and, if so, how.
A:
It is no set.
We start with «1» which can be thought of as the empty set. Then we add $1$ to the empty set. Then we add $1/2$ to the result. The new result can be thought of as the empty set.
Then we add $1/2$ to the result. The result can be thought of as the empty set.
Then we add $1/3$ to the result. The result can be thought of as the empty set.
Continue inductively, the new result can be thought of as the empty set.
The result is the empty set for some infinite number of steps, but always the final result is the empty set.
The result does not accumulate because the empty set is the empty set at every step. No matter how far up the process you go, you will never go beyond the empty set, no matter how many steps you do.
A different way to prove it:
Wlog we have $0 le frac{1}{2}, frac{1}{3}, frac{1}{4}, dots$
We have:
$f_1 = 1
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