# Density Theorem and Limit Proof

I'm new to proof writing and have trouble understanding hoe to structure or get proofs started. An example proof that my teacher left as practice is below. Any suggestions? Of course, I know I must use the Density Theorem I am just not sure how.

Prove that given x ∈ ℝ there is a sequence r ∈ ℚ such that r -> ℚ as n -> ∞.

• Your teacher's practice problem was " Prove that give $x\in\mathbb{R}$ there exists a sequence $r_n\in\mathbb{Q}$ such that $r_n\rightarrow x$ as $n\rightarrow\infty$". Do you understand the differences? – Yanko Oct 17 '17 at 13:05
• @yanko To be more precise, you should probably write a sequence $(r_n)\subset \mathbb Q$ such that $r_n \to x$ as $n \to \infty.$ – Epiousios Oct 17 '17 at 13:11
• @Liza are you allowed to use that in every $\varepsilon$-neighborhood of $x$ there is a rational number? – Epiousios Oct 17 '17 at 13:13
• Oh yes, I do understand why it needs to be rn. Yes I believe so @Epiousios – Liza Oct 17 '17 at 18:46

I'll assume that you are allowed to use that in every $\varepsilon$-neighborhood of a real number there are rational numbers.
Let $x\in \mathbb R$. Because $\mathbb Q$ is dense in $\mathbb R,$ for any $n\in \mathbb N$ we can find $r_n\in \mathbb Q$ such that $|r_n-x|<1/n$. Then for any $n,$ $$0\le|r_n-x|\le 1/n,$$ so that $|r_n-x|\to 0,$ as $n\to\infty$ by the Squeeze Theorem. That means that $r_n\to x,$ as $n\to \infty.$
• The $1/n$ is just a convenient sequence that converges to zero. You could also take $1/n^2$ or $2^{-n}.$ – Epiousios Oct 17 '17 at 20:57