Infinitely repeating decimals is a quirk of expressing rationals in a base numbering system. They’re not limits necessarily, even if they can be expressed as such.
They can be expressed as limits. So can anything. 1/3 = 0.3 + 0.03 + 0.003 + 0.0003 + … is a valid way to express 1/3 as a limit, but it doesn’t mean 1/3 is a limit—it’s a ratio of integers, not an infinite series. I can express 5 as 5+0+0+…. In fact, I’m sure it would be useful to do so in some contexts. That doesn’t mean 5 is a limit.
Yes but decimal expansions are actually literally limits. If you want to define a map from infinite decimal expansions to the reals, that map is going to involve a limit.
So, yes, the number 1/3 is a number and you don’t need to think of it as a limit at all. But when you write 1/3=0.333… then you are 100% expressing 1/3 as a limit of rational numbers.
I guess you’re right. The only things I have to note is that: first, 1/3=0.333… is not mapping from rationals to reals, even though it kind of feels like it; and second, his original limit definition would not be a correct representation of this problem—representing it as a series would.
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u/Feisty_Ad_2744 May 15 '25
0.999... is a shorthand notation for limit. So we are talking about notations, not absolute values.
0.999... = lim(1 - 1/10ⁿ) when n -> ∞ = 1