The problem appears in Tom Apostol's Calculus, Volume 1, pages 31,32. This is introductory material, related to foundations of real number system.

Quote:

No proof is given and as often happens to me, contrary to the claim, I could not do the thing easily.Since we shall do very little with decimals in this book, we shall not develop their properties in any further detail except to mention how decimal expansions may be defined analytically with the help of the Least-Upper-Bound axiom.

If x is a given positive real number, let denote the largest integer Having chosen , we let denote the largest integer such that

More generally, having chosen we let denote the largest integer such that

Let S denote the set of all numbers

obtained this way for n=0,1,2... Then S is nonempty and bounded above, and it iseasy to verify that x is actually the least upper bound of S.

So I need a proof that sup(S)=x.

For this purpose I should use Least-Upper-Bound axiom which roughly states that any set of real numbers that has upper bound has supremum too (which is real number).

I did try to prove it, resulting somewhat ugly and cumbersome proof, which most surely has weaknesses. I'm giving it here in hope someone could

comment on it. My attempt at the proof uses two intermediate facts.

a/If a,x,y are real numbers which satisfy equalities for any then .It is proven in the textbook as consequence of the Least-Upper-Bound axiom

b/for any .

The proof is by induction on n. Case n=1 is obviously true, so assuming

is true we need to prove

Since

if we manage to prove that

we shall have

.

This is easy

which is true for all .

Now to the proof of the main statement.

By construction of x,

for any and x is an upper bound of S.

By Least-Upper-Bound axiom, S has supremum and it's a real number. Denote . Then

for any .

Combining the two last inequalities with b/ we can conclude that

for any .

And because y is least upper bound we have

for any positive integer n. Applying a/ gives us the equality x=y.

I'd appreciate any comments as well as one-liner proof by contradiction for example.

Live long and prosper ;-).