0.999... = 1 (?)

Just like there's proof of 0.999... = 1, there's proof that 0.999... =/= 1.

Take these two definitions for example:

1 = lim (1 + 1/x) as x approaches infinity
1 = lim (1 - 1/x) as x approaches infinity

Then:

lim (1 + 1/x) as x approaches infinity - lim (1 - 1/x) as x approaches infinity
= lim (2/x as x) approaches infinity
= 0.

This is generally accepted by mathematicians.

But what happens when you repeat this process infinite times? (This is elaborated by my post below.)
What you end up with is the following:

1 = lim (1 - x/x) as x approaches infinity
1 = lim ((x-x)/x) as x approaches infinity
1 = lim (0/x) as x approaches infinity
1 = 0 ?

Since that is impossible, 1/x accounts for something, and 1 =/= 0.999...


I voted for yes by the way.

See this:


This is all nice and dandy, but, repeating adding an "infinitesimal" value infinite times still equals zero. This is because you have to use limits to calculate for infinity.

This is the "infinitesimal" value we're going to add an infinite times. This can be done two ways, either by summing infinite times, or by multiplying by infinity:


And then you thus have to evaluate the inner limit before evaluating the outer one.

You can also think of it this way: If the "infinitesimal" value counts for something, why is it true that what you ended up with became zero, and not negative infinity? That kind of doesn't make sense, right?

Noah