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## Main Question or Discussion Point

In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that

"for any irrational number there exists a sequence of rational numbers that converges to it",

and it doesn't have a proof for it, just saying that it is a consequence of the fact that ##\mathbb{Q}## is dense in ##\mathbb R##.

Any ideas how to proceed for a rigorous proof?

My idea is that if ##x=x_0.x_1x_2...x_n,...## is the representation of the irrational x in the decimal system with ##x_i \in {0...9}## then the sequence

##\sigma_n=\sum\limits_{k=0}^{n}\frac{x_k}{10^k}## is rational and converges to the number but something tells me this is not a rigorous proof.

"for any irrational number there exists a sequence of rational numbers that converges to it",

and it doesn't have a proof for it, just saying that it is a consequence of the fact that ##\mathbb{Q}## is dense in ##\mathbb R##.

Any ideas how to proceed for a rigorous proof?

My idea is that if ##x=x_0.x_1x_2...x_n,...## is the representation of the irrational x in the decimal system with ##x_i \in {0...9}## then the sequence

##\sigma_n=\sum\limits_{k=0}^{n}\frac{x_k}{10^k}## is rational and converges to the number but something tells me this is not a rigorous proof.

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