From: Westbrick | #067
Please explain to me, in words, why "infinitely close to" means "is the same as." We all understand that the proofs work out this way, but, as has been shown time and time again in this topic, anything involving infinity is going to be incredibly hard to wrap one's head around.
Sure, so you just want more detail in the proof to be convinced. That is fine, and isn't an indication of stupidity.
Well here's the best way to explain it. Think of any 2 different real numbers (decimal numbers). Now think of a number in between those 2 numbers. It's ALWAYS possible, right?
Now think of 2 equal real numbers, 2.5 and 2.5 or something. Try to think of a number between those 2 numbers. Impossible, right?
Now try to think of a number between .999 repeating and 1. Impossible, right? That's because .999 repeating is infinitely close to 1, so there does not exist a number between it and 1. Therefore they are equal, and that's why infinitely close is equivalent to equality.
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_foolmo_
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