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Formally, I want to prove that if $x$ and $y$ are real numbers such that $x \lt y$ then there exists a real number $z$ such that $x \lt z \lt y$.

I want to know whether, in constructing the proof, I should think in terms of continuity of the real line or not. In other words, is the proof of the existence of a real between any two reals equivalent to proving the real line is continuous?

This is a question in a calculus textbook I am going through as part of a self study project to brush up on my calculus. The text is called Calculus Volume 1 written by Tom M. Apostol page 19 exercise 1.14*.

MHall
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  • Can you prove that "if x and y are real numbers then x+ y is a real number" (that the real numbers are closed under addition)? Can you prove then that (x+ y)/2 is a real number? Do you see that "if x< y then 2x< x+ y? And then that x< (x+ y)? What if y< x? – user247327 Dec 28 '18 at 21:52
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    By "the real line is continuous" I assume you mean something (informally) along the lines of "there are no gaps", in which case, this is because $\Bbb R$ is a complete metric space. – Edward Evans Dec 28 '18 at 21:53
  • i would ask, what is the definition of "less than"? is it defined by subtraction? if so, then how do you define subtraction of two real numbers? if i give you any two real numbers, how can you subtract them and what is the result? as someone who leans towards finitism, i must confess i believe this is equivalent to asking if there is a number between two infinite numbers, but i am in a tiny minority. the interesting thing to me is that the operation of subtraction has a different relationship to the infinites than it does to the finites, in terms of output type and input type. – don bright Dec 29 '18 at 15:35

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First, let me say a bit about the end of your question. I'm not sure what you mean by "the real line is continuous," but I suspect you mean that the real line is connected or complete. If so, then no, that's a different thing; for example, $\mathbb{Q}$ has the "density" property here (between any two rationals, there's another rational) but is neither connected (consider $(-\infty,\pi)$ versus $(\pi,\infty)$, for example) or complete (consider the sequence $3,3.1,3.14,3.141,...$, for example).

Since connectedness/completeness can be a bit technical at first, let me say the following: intuitively, a space is connected if it doesn't "break into a bunch of separate pieces," and is complete if it "doesn't have any gaps" - the set $\mathbb{Q}$ of rationals, even though it's dense, has lots of gaps and breaks apart really easily.

As to proving the claim you're looking at, I suspect you're thinking too hard about it. First, try to guess what a good formula for such a $z$ might be, and then try to prove (using whatever formal system you've been given) that it actually works. And this formula will be quite simple.

HINT: if on a quiz Sam scored $10/10$ and Alex scored $6/10$, then together they got an ---- score of $8/10$. Now, how do you calculate the "---" of two numbers?

Noah Schweber
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