Rationals are Everywhere Dense in Reals/Normed Vector Space

Theorem

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.

Let $\Q$ be the set of rational numbers.

Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$

Proof

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We have that Between two Real Numbers exists Rational Number:

$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$

Let $a := x$ with $x \in \R$.

Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.

Let $b := x + \epsilon$.

Then:

\(\ds x - \epsilon\)

\(<\)

\(\ds x\)

\(\ds \)

\(<\)

\(\ds r\)

\(\ds \)

\(<\)

\(\ds x + \epsilon\)

Since this holds for all $x$, we have that:

$\forall x \in \R : \forall \epsilon \in \R_{\mathop > 0} : \exists r \in \Q : \size {x - r} < \epsilon$

By definition, $\Q$ is dense in $\R$.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces