Why do you think the sum 10/11+10/121+10/1331+... is greater than 9/10+9/100+9/1000+..., even hypothetically?

Well, it's because 10/11+10/121+10/1331 approaches 1 more quickly than 9/10+9/100+9/1000.

Suppose we were to graph y_{11}=((11^{x})-1)/(11^{x}) and y_{10}=((10^{x})-1)/(10^{x}). Regardless of which x you choose, y_{11} is always greater than y_{10}. So it stands to reason that hypothetically, if y_{11} and y_{10} were single values rather than functions, y_{11} would be greater than y_{10}.

In fact, every base has a similar function/number. But no matter which base you choose, none of them will ever equal 1.

Objection! There is a major hole in my theory.

y

_{10}=((10

^{x})-1)/(10

^{x})

y

_{11}=((11

^{x})-1)/(11

^{x})

y

_{10} < y

_{11}True so far. However, this number "0.999..." is ambiguous. At first glance it appears to represent 9/10 + 9/100 + 9/1000 + ... (after all, there is a 9 in the tenths place, a 9 in the hundredths place, a 9 in the thousandths place, and so on). And if that be the case, it's the same thing as y

_{10}.

But who says that that has to be the case? Couldn't it just as well represent 99/100 + 99/10000 + 99/1000000+...? (After all, ignoring the carry, there is a 99 in the hundredths place, a 99 in the ten-thousandths place, and so on). And if that be true then 0.999... represents not y

_{10} but y

_{100}:

y

_{100}=((100

^{x})-1)/(100

^{x}), which approaches 1 much more quickly than y

_{10} or even y

_{11} for that matter.

(Come to think of it, 0.999... could represent any of functions y

_{10n}.)

Same for base-eleven's "0.AAA...". No way to know whether it represents y

_{11} or y

_{121} or y

_{11n}.

And y

_{121} > y

_{100} > y

_{11} > y

_{10}. So it's like, yeah, y

_{11} is greater than y

_{10}, but it's sandwiched between y

_{10} and y

_{100}, which itself is sandwiched between y

_{11} and y

_{121}, making it impossible to know for sure that 0.999... is lesser than 0.AAA..., unless of course we ignore all bases which are powers of other bases. But why anyone would feel so inclined to do that remains a mystery.