>>12380111He is using the metric space consisting of the rational numbers with the distance metric d(a,b) = |arctan(a) - arctan(b)|. He is defining the sequence f_n = n, i.e. [0, 1, 2, 3, 4, ... ]. That is, the natural numbers in the usual order.
The expression arctan(x) gets increasingly close to pi/2 as x gets closer to infinity. Or to be more precise about it, for the regular real numbers, the limit of arctan(x) as x tends to infinity is pi/2. That means that for two very large natural numbers x and y, arctan(x) and arctan(y) are both very close to pi/2, which means that |arctan(x) - arctan(y)| is close to zero. For example, |arctan(10000) - arctan(1000)| is about 0.0009, and |arctan(1000000) - arctan(100000)| is about 9*10^-6.
In particular, for all epsilon > 0, there is an N, such that for all natural a,b>N, |arctan(a) - arctan(b)| = d(a, b) = d(f_a, f_b) < epsilon. This means that f is a Cauchy sequence *in the metric space defined by the metric d*.
But there is no rational number Q that is the limit of the sequence f under this metric. This means that f does not converge to any point in the metric space defined here, AKA does not converge, period.
If you squint, you could say that f has a limit at infinity, and f converges to this point outside the metric space. But that doesn't actually mean anything, because the distance metric d() isn't even defined for infinity, it is defined only for rational numbers. For this reason, the notion of "this sequence converges to something, but that something is not in the metric space" is bunk. Convergence is meaningful only relative to the space it's in, because that's the only place where the distance metric is defined. For that reason, the standard terminology here is that "f does not converge", full stop.
The sequence f does do something kinda like converging, and that something is "being a Cauchy sequence". Which is why "Cauchy sequence" and "convergent sequence" are two different things.
