Q&A: How To Explain Why Two Numbers with Arbitrarily Small Distance Are Equal

Cross-posted from here.

How can I explain the following to students who are struggling to grasp it?

$$\begin{align*} |a – b| < \epsilon, \forall \epsilon > 0 \iff a = b \end{align*}$$


I’d recommend to play a little game with them. You choose a number $a.$ They choose a number $b$ that they think will satisfy $\vert a-b \vert < \epsilon \,\,\, \forall \epsilon > 0.$

You try to state a counterexample, that is, an $\epsilon > 0$ such that $\vert a-b \vert \geq \epsilon.$ If you can do this, you win. Otherwise, if you can’t, then they win.

Enough rounds of this, and they should see that the only way they can win is if they choose $b = a.$

Some additional notes:

  • To help students develop the right intuition, I'd recommend to refer to $|a-b|$ as the distance between $a$ and $b$ throughout the game. (And if they don't understand why it represents distance, then it would be worth pausing the game to review that, since it's such an important part of the intuition.)
  • This game not only leads the students to intuitively grasp the equivalence, but also leads them into the proof: after enough rounds, they should also see that there's a simple rule for producing some $\epsilon$ for each $b.$