Proof verification that the sequence $x_n = \frac{1}{n}$ converges to every point of $\mathbb{R}$ on the cofinite topology

Your argument is fine, but you are right that such specifics are unnecessary. Any sequence that does not contain a particular point infinitely many times converges to every point in an infinite space with the cofinite topology.

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General Topology

Real Analysis

Proof Verification

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