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-

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

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

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