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There is another proof on math stack exchange, but I would like to ask if the following reasoning makes any sense:

It contains all of its limit points, namely the "singleton"(?) sequence, the sequence with only one element, namely the element in the singleton. Or, is this not in line with the definition of a sequence? Must a sequence have infinitely many elements?

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Your reasoning is correct, but the justification should be rephrased. Rather than a sequence with one element, you should instead be referring to a constant sequence. A sequence is not a set of elements but a ''list of points'' given by a function $x_n : \mathbb{N} \to X$

With this terminology, you should say that the only sequence in $\{x\}$ is the constant sequence $(x_n)$ with $x_n = x$ for all $n$. Thus, the limit of every convergent sequence in $\{x\}$ is an element of $\{x\}$.

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