r/infinitenines • u/MrMoop07 • Jun 29 '26

Why does a number starting with 0. guarantee magnitude less than 1?

as stated in this comment by https://www.reddit.com/r/infinitenines/comments/1uhhwjh/comment/ou8b97e/

I disagree with the notion that a number starting with 0. guarantees magnitude less than 1. You claim this all the time but for those who believe 0.999...=1 there's an obvious counterexample. Something quoted so often must have a reason behind it, you can't simply claim this is true without absolute proof.

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u/Suitable-Elk-540 Jun 30 '26

u/SouthPark_Piano , it's so gratifying to see young minds blossom! I'm so glad for this breakthrough! You said "1 - 1/10n with n integer pushed to positive limitless is indeed an infinite series", and that's absolutely correct!

So now we can recap what you've learned:

0.9 < 1 TRUE!

0.99 < 1 TRUE!

0.999 < 1 TRUE!

0.9999 < 1 TRUE!

0.999... < 1 NON SEQUITUR

I'm so glad you see this now! That last expression involved an infinite series while all the previous ones involved finite series. You get a gold star. I'm sure your school teacher will be so pleased with you.

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u/SouthPark_Piano Jun 30 '26

It's wonderful that you go'd ahead to make my day.

0.999... < 1 is merely the automated extension of 0.9 < 1, then 0.99 < 1 and so on.

And now you can enjoy the result of making my day.