Limit points are points contained in the closure of every end piece

Let $𝑋$ be a topological space, $(𝑥 𝑛) \infty 𝑛 = 1 \in 𝑋$ a sequence, and $𝑀 𝑚 = {𝑥 𝑛} \infty 𝑛 = 𝑚$ denote an end piece. Then $𝑎 \in 𝑋$ is a Limit point of $𝑥 𝑛$ iff $𝑎 \in C l 𝑀 𝑛$ (closure) for all $𝑛 \in ℕ$ . topology

Proof

proof See @looseAlgebraischeTopologie2010, p. 19

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Limit points are points contained in the closure of every end piece

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