EDB — 13W

[13W]Let \((X,𝜏)\) be a topological space and \(f:X→ℝ\) a function. Let \(\overline x∈ X\) be an accumulation point. Let eventually \(U_ n\) be a family of open neighbourhoods of \(\overline x\) with \(U_ n⊇ U_{n+1}\). Then there exists a sequence \((x_ n)⊂ X\) with \(x_ n∈ U_ n\) and \(x_ n≠ \overline x\) and such that

(Note that in general we do not claim neither expect that \(x_ n→\overline x\)).