First-countable implies compactly generated

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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Statement

Property-theoretic statement

The property of topological spaces of being first-countable is stronger than the property of being compactly generated.

Verbal statement

Any first-countable space is compactly generated.

Proof

Given: A first-countable space $X$

To prove: There exists a collection of compact subsets $\{K_{i}\}_{i\in I}$ of $X$, such that $U\subset X$ is open if and only if $U\cap K_{i}$ is open in $K_{i}$ for every $i$

Construction of the collection of compact subsets

We consider compact subsets of the following form: take a sequence $x_{n}$ that converges to a point $x$ (note that a sequence may converge to more than one point, we just make sure there is at least one point of convergence). Now define the compact set $K$ corresponding to this sequence as $\{x_{n}\}_{n\in \mathbb {N} }\cup \{x\}$.

Let's first see why $K$ is compact. In