----- > [!proposition] Proposition. ([[the rationals are dense in the reals]]) > For any $x \in \mathbb{R}$, for any $\varepsilon>0$, there exists $y \in \mathbb{Q} \cap B_{\varepsilon}(x)$. > [!proof]- Proof. ([[the rationals are dense in the reals]]) > Recall that every [[real numbers|real number]] is the limit of rational numbers. Fixing $x \in \mathbb{R}$ and $\varepsilon>0$, obtain a [[sequence]] $(x_{n})$ of rational numbers [[converge|converging]] to $x$. Then obtain $N \in \mathbb{N}$ s.t. for all $n > N$ we have $x_{n} \in B_{\varepsilon}(x)$. Since by construction $x_{n} \in \mathbb{Q}$, we have $x_{n}:=y \in \mathbb{Q} \cap B_{\varepsilon}(x)$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```