Prove-using-the-density-of-Q-in-R-that-every-real-number-x-is-the-limit-of-a-cauchy-sequence-of-rational-numbers-r-n-n-N-Give-a-sequence-of-irrational-numbers-S-n-such-that-S-n-x-

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Question Number 11799 by tawa last updated on 01/Apr/17

$$\mathrm{Prove}\mathrm{using}\mathrm{the}\mathrm{density}\mathrm{of}\mathbf{Q}\mathrm{in}\mathbb{R}\mathrm{that}\mathrm{every}\mathrm{real}\mathrm{number}\mathrm{x}\mathrm{is}\mathrm{the}\mathrm{limit}\mathrm{of}$$$$\mathrm{a}\mathrm{cauchy}\mathrm{sequence}\mathrm{of}\mathrm{rational}\mathrm{numbers}{\left({\mathrm{r}}_{\mathrm{n}}\right)}_{\mathrm{n}\in \mathrm{N}}.\mathrm{Give}\mathrm{a}\mathrm{sequence}\mathrm{of}\mathrm{irrational}$$$$\mathrm{numbers}\left({\mathrm{S}}_{\mathrm{n}}\right)\mathrm{such}\mathrm{that}{\mathrm{S}}_{\mathrm{n}}\to \mathrm{x}.$$

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