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 -> ∞.

  • $\begingroup$ 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? $\endgroup$ – Yanko Oct 17 '17 at 13:05
  • $\begingroup$ @yanko To be more precise, you should probably write a sequence $(r_n)\subset \mathbb Q$ such that $r_n \to x$ as $n \to \infty.$ $\endgroup$ – Epiousios Oct 17 '17 at 13:11
  • $\begingroup$ @Liza are you allowed to use that in every $\varepsilon$-neighborhood of $x$ there is a rational number? $\endgroup$ – Epiousios Oct 17 '17 at 13:13
  • $\begingroup$ Oh yes, I do understand why it needs to be rn. Yes I believe so @Epiousios $\endgroup$ – 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 $n\to \infty.$

  • $\begingroup$ I understand most of that. Except, where does the 1/n come from? $\endgroup$ – Liza Oct 17 '17 at 18:49
  • $\begingroup$ The $1/n$ is just a convenient sequence that converges to zero. You could also take $1/n^2$ or $2^{-n}.$ $\endgroup$ – Epiousios Oct 17 '17 at 20:57

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