How does the sum of 1/2 + 1/4 + 1/8 etc. (Halves each time) eventually become 1? I understand it’s infinite but surely it still wouldn’t work?
In: Mathematics
The key here is the Archimedean property of real numbers which states the following: for any positive number x, however small, there exists a large enough natural number n such that x>1/n.
Imagine you subtract 1/2, then 1/4, then 1/8 and so on from 1. If we assume there is a number x we are left with after this process finishes, the number x is less than 1/2, less than 1/4, less than 1/8 and so on.
That means I can show x is simultaneously less than any number 1/n and therefore it’s not *positive*. Hence it is zero. (I don’t think I need to convince you it’s not negative).
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