Actually, it’s fairly trivial to prove that between any two distinct real numbers there is a number whose decimal expansion contains every finite sequence of decimal digits: first construct one such number (necessarily between 0 and 1) as follows:
Concatenate all one digit strings, followed by the concatenation of all 2 digit strings, followed by the concatenation of all 3 digit strings, etc. with the strings of given length listed in lexicographic order with respect to the usual order on the digits
(the number’s decimal expansion begins
.0123456789000102030405060708091011121314151617181920212223... )
Now it’s just a matter of multiplying this by a small enough power of 10 and adding some digits at the front to get it in between the two given numbers. I leave the details as an exercise.
I think, however, that what you are intuitively grasping for is the assertion that
almost every real number has a decimal expansion in which every finite sequence of decimal digits occurs — where “almost every” had the technical meaning of the complementary set being of measure zero (can be contained in a union of intervals the sum of whose lengths may be chosen to be less than any given positive number).
I think I can see a route to proving this: since the number of finite decimal strings is countable and countable unions of sets of measure zero are of measure zero, it actually suffices to show that for every finite decimal string, the set of real numbers whose decimal expansion does not contain it is of measure zero. And I think the set you get by removing the places where the given string occurs in the decimal expansion will end up being a (subset of a) Cantor set, and thus of measure zero (but it’s getting late here and I’m not going to try to set down a rigorous proof tonight).
By the way — if it’s a foregone conclusion logically, then there’s a mathematical proof, and conversely. The only things which are foregone conclusions logically are mathematical theorems.
“I think, however, that what you are intuitively grasping for is the assertion that
almost every real number has a decimal expansion in which every finite sequence of decimal digits occurs — where “almost every” had the technical meaning of the complementary set being of measure zero (can be contained in a union of intervals the sum of whose lengths may be chosen to be less than any given positive number).”
Yes, that’s probably what I was getting at, but not having had to engage in any real mathematical arguments for many years, I’m too rusty to get my thoughts across as well as you :)
“By the way — if it’s a foregone conclusion logically, then there’s a mathematical proof, and conversely. The only things which are foregone conclusions logically are mathematical theorems.”
Yes, I should have said a foregone conclusion intuitively, which is often, but not always an indicator that a mathematical proof can be found. For example, when I used to solve polynomials day in and day out, I could intuitively know what the solutions would be without having done the “grunt work” of solving them by the proper means. Most of the time, once I did the work, my intuition was right. Our unconscious minds are constantly doing complex math intuitively that most people struggle with understanding consciously. So, I think that accounts for why some tricky mathematical problems seem to have intuitively true solutions, even if mathematicians can’t figure out the exact proofs yet.