Hearted Youtube comments on Mathologer (@Mathologer) channel.
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A relatively easy proof:
Any time you finish adding positive numbers (i.e. any time you go above the limit), it doesn't do any harm to add up to a fixed number of extra positive numbers. Let's say add an arbitrary amount of extra positive numbers that is between 0 and 9.
Because the sequence of positive numbers converges to 0, the distance this gets you away from the target real number will be arbitrarily small (namely, in total 10 times the epsilon from the definition of being a zero-sequence), and so, if you do this, your rearrangement will still converge to the same real number.
But this means that any real number R on [0, 1) can be turned into a new rearrangement:
Take the decimal expansion of R. Then when creating your rearrangement and deciding how many extra positive numbers you want to add at the end of one consecutive sequence of positive numbers, just choose the next decimal digit of R, and add as many extra positive numbers as that digit is high.
Because the set of real numbers on [0, 1) is uncountably infinite, and each number gives a distinct rearrangement, the set of all rearrangements resulting in the same sum must also be uncountable.
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