Cauchy sequence
Non-example: open interval

The open interval X = (0, 2) {\displaystyle X=(0,2)} in the set of real numbers with an ordinary distance in R {\displaystyle \mathbb {R} } is not a complete space: there is a sequence x n = 1 / n {\displaystyle x_{n}=1/n} in it, which is Cauchy (for arbitrarily small distance bound d > 0 {\displaystyle d>0} all terms x n {\displaystyle x_{n}} of n > 1 / d {\displaystyle n>1/d} fit in the (0, d) {\displaystyle (0,d)} interval), however does not converge in X {\displaystyle X} — its 'limit', number 0, does not belong to the space X. {\displaystyle X.}
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