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I attempting to solve the following question: "Show there is a sequence of rational numbers converging to any irrational number".

I don't understand how to tackle this problem because it says any irrational number.

I know for example that the sequence $s_n$ = $(1 + 1/n)^n$ converges to $e$, but I can't think of a sequence that will converge to any rational number.

As a caveat I have looked at other similar questions on stack exchange but none seem to cover this any aspect.

Thank you.

Bob
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    show that for every irrational number there is a rational number as close as you like. Use the Archimedean property for this. – Sarvesh Ravichandran Iyer Apr 20 '16 at 11:07
  • @астонвіллаолофмэллбэрг how would I do this? – Bob Apr 20 '16 at 11:10
  • An intuitive way to think about this, is to imagine representing your irrational number in decimal form, and then making the sequence out of the first $n$ digits of that representation (the denominator will increase by a factor of $10$ each time). This will give a sequence of rationals converging to the desired irrational. (Note that this is not quite enough to formally prove it, but the idea is a helpful intuition.) – Justin Benfield Apr 20 '16 at 11:10
  • I find the best way to avoid the notion of $\mathbb Q$ dense in $\mathbb R$ is the comment of Justin Benfield. You just need taking in account that the decimal form of irrational never ends. – Piquito Apr 20 '16 at 11:39

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The space of real numbers $R$ is the metric completion of $Q$ the space of rational numbers. To construct $R$, start with $Q$, and let $C$ be the space of Cauchy sequences of $Q$. Two elements $(x_n)$ and $(y_n)$ are equivalent if and only if $(x_n-y_n)$ converges towards $0$, then $R$ is the quotient space for this equivalence relation. You can identify $Q$ with the equivalence classes of constant sequences.

If $[x_n]$ is the class of the Cauchy sequence $(x_n)$, define $(y_n^i)$ by $y^i_l=x_i$ for every $l$. Then $([y^i_n])$ converges towards $[x_n]$ and $[y_n^i]$ can be identified with the rational $x_i$

Tsemo Aristide
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