A proof of the density of the rational numbers in the real numbers:

Take any \(a\) and \(b\) in the real numbers such that \(a < b\). Then \((b-a) > 0\).

By the Archimedean property of the real numbers, there exists some natural number \(n\) such that \(n > \frac{1}{b-a}\)

\(n(b-a) > 1\)

\(nb-na > 1\)

Because the difference between \(na\) and \(nb\) is greater than \(1\), there must exist at least one integer \(k\) between them:

\(na < k < nb\)

\(a < \frac{k}{n} < b\)

\(\frac{k}{n}\) is a rational number because \(k\) and \(n\) are both integers

Thus, between any two real numbers, \(a\) and \(b\) there exists a rational number.

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A proof of the density of the irrational numbers in the real numbers:

Take any \(a\) and \(b\) in the real numbers such that \(a < b\)

\(\sqrt{2} > 0\), so \(\frac{a}{\sqrt{2}} < \frac{b}{\sqrt{2}}\)

\(\frac{a}{\sqrt{2}}\) and \(\frac{b}{\sqrt{2}}\) are real numbers, so by the density of the rationals in the real numbers, there exists some rational number \(q\) between \(\frac{a}{\sqrt{2}}\) and \(\frac{b}{\sqrt{2}}\)

(Note: if \(q\) would be \(0\), then instead find some \(q\) between \(\frac{a}{\sqrt{2}}\) and \(0\).)

\(\frac{a}{\sqrt{2}} < q < \frac{b}{\sqrt{2}}\)

\(a < \frac{q}{\sqrt{2}} < b\)

\(\frac{q}{\sqrt{2}}\) is irrational (if \(\frac{q}{\sqrt{2}} = r\), where \(r\) is rational, then \(\sqrt{2} = \frac{r}{q}\), where \(\frac{r}{q}\) is rational, and thus \(\sqrt{2}\) would be rational, which is false, as \(\sqrt{2}\) is irrational)

Thus, between any two real numbers, \(a\) and \(b\), there exists an irrational number.

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The rational and irrational numbers, \(\frac{k}{n}\) and \(q\sqrt{2}\), found between any two real numbers are themselves real numbers. In finding them, new intervals are created wherein can be found additional rational and irrational numbers. With each successive exploration, the well of numbers between any two real numbers deepens.

What hides in the small spaces of your life? Where are your infinite reservoirs of strength, love, inspiration, wonder, ...? Finding an insight doesn't have to be the end of the journey. Use that insight to open up a new depth and push further. The choice of \(a\) and \(b\) is arbitrary; two people may begin their explorations from different starting points, finding entirely disparate lines of insight, but each explored depth can be equally infinite and rich in its truths.

Take any \(a\) and \(b\) in the real numbers such that \(a < b\). Then \((b-a) > 0\).

By the Archimedean property of the real numbers, there exists some natural number \(n\) such that \(n > \frac{1}{b-a}\)

\(n(b-a) > 1\)

\(nb-na > 1\)

Because the difference between \(na\) and \(nb\) is greater than \(1\), there must exist at least one integer \(k\) between them:

\(na < k < nb\)

\(a < \frac{k}{n} < b\)

\(\frac{k}{n}\) is a rational number because \(k\) and \(n\) are both integers

Thus, between any two real numbers, \(a\) and \(b\) there exists a rational number.

- - - - - - - - - -

A proof of the density of the irrational numbers in the real numbers:

Take any \(a\) and \(b\) in the real numbers such that \(a < b\)

\(\sqrt{2} > 0\), so \(\frac{a}{\sqrt{2}} < \frac{b}{\sqrt{2}}\)

\(\frac{a}{\sqrt{2}}\) and \(\frac{b}{\sqrt{2}}\) are real numbers, so by the density of the rationals in the real numbers, there exists some rational number \(q\) between \(\frac{a}{\sqrt{2}}\) and \(\frac{b}{\sqrt{2}}\)

(Note: if \(q\) would be \(0\), then instead find some \(q\) between \(\frac{a}{\sqrt{2}}\) and \(0\).)

\(\frac{a}{\sqrt{2}} < q < \frac{b}{\sqrt{2}}\)

\(a < \frac{q}{\sqrt{2}} < b\)

\(\frac{q}{\sqrt{2}}\) is irrational (if \(\frac{q}{\sqrt{2}} = r\), where \(r\) is rational, then \(\sqrt{2} = \frac{r}{q}\), where \(\frac{r}{q}\) is rational, and thus \(\sqrt{2}\) would be rational, which is false, as \(\sqrt{2}\) is irrational)

Thus, between any two real numbers, \(a\) and \(b\), there exists an irrational number.

- - - - - - - - - -

The rational and irrational numbers, \(\frac{k}{n}\) and \(q\sqrt{2}\), found between any two real numbers are themselves real numbers. In finding them, new intervals are created wherein can be found additional rational and irrational numbers. With each successive exploration, the well of numbers between any two real numbers deepens.

What hides in the small spaces of your life? Where are your infinite reservoirs of strength, love, inspiration, wonder, ...? Finding an insight doesn't have to be the end of the journey. Use that insight to open up a new depth and push further. The choice of \(a\) and \(b\) is arbitrary; two people may begin their explorations from different starting points, finding entirely disparate lines of insight, but each explored depth can be equally infinite and rich in its truths.