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User:ConMan/Proof that 0.999... equals 1 (Limit proof)

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The following proof was originally posted in Talk:Proof that 0.999... equals 1/Archive05, and I am keeping it here for my, and others', future reference.

Motivation

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A number of anonymous posters on the Talk page claimed that while and , the two were not equal because "the infinite sum is not equal to the limit of the partial sums". I seemed to recall something about the infinite sum being defined as the limit of partial sums, because nothing else makes sense, but I wondered if it was in fact provable - and in this case at least it was.

Accepted definitions and statements

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These were agreed upon by people claiming both that 0.999... equals and does not equal 1.

  1. , and in particular, such that .
  2. Given a sequence , means (ie. is defined as) such that

Not true. I was the other debater in this argument with Rasmus. His assertions are false and what you have in the archive is no proof at all. Here are a few recent articles I wrote on this subject:

There is a problem defining the sum of an infinite sum as a limit: http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/magnitude_and_number.pdf

And, 0.999... is not really a number: http://thenewcalculus.weebly.com/uploads/5/6/7/4/5674177/proof_that_0.999_not_equal_1.pdf

The proof

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Let .

by point #3.

, where M is some finite number greater than (which exists by point #2).

For any given , set . Then:

Therefore, we now have that:

, and since , we then know that .

By point #4, . It has already been agreed that , and therefore . In other words, I have shown that, in fact, that "the infinite sum is equal to the limit of the partial sums" is not a definition, but a provable statement.

No. It has not been agreed anywhere that , this is what you are trying to prove.

Holes

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I admitted that the proof as stated above was not 100% rigorous, so here is a list of some of the spots where the rigor is lacking, and an attempt to correct that.

Two limits?

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As pointed out by User:Rasmus Faber, the proof assumes that if a sequence has a limit, that limit is unique. In the real numbers, this can be shown by the following Lemma:

Lemma 1: If a sequence of real numbers converges, it has a unique limit.

Proof:

Suppose that the sequence is convergent. Then is not undefined.

Assume that and , but that . Then by the definition of the limit:

such that , and similarly with a replaced by b.

Now, by the Triangle Inequality. However, the fact that a and b are both limits of the means that for , say, there are values of n for which , a clear contradiction. Therefore .