They're not right if they are talking about the real numbers. But they could be right in some other number system, like the hyperreal numbers! In the hyperreal numbers, there's a number smaller than 1 and bigger than
0.9
and bigger than
0.99
and bigger than
0.999
and so on. You can call this number 0.9999..., and then
0.9999... < 1
But you have to be careful about that "..." thing. The hyperreals also contain different sizes of infinite numbers! So, you have to say how many 9's you mean: just saying there are infinitely many is too vague.
If you let the 9's go on infinitely *for the smallest size of infinity*, but then stop and put 0's after that, then you get a number smaller than 1.
In fact you can stop after any size of infinity, and you'll get a number smaller than 1. But you get different numbers this way.
And if you go on *forever*, you get 1.
All this is explained here:
• Karin Usadi Katz and Mikhail G. Katz, When is .999... less than 1?, The Mathematical Enthusiast 7 (2010), 3–30, https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1381&context=tme
• A. Harold Lightstone, Infinitesimals, American Mathematical Monthly 79 (1972), 242–251, https://www.jstor.org/stable/2316619?seq=1
Thanks to Oscar Cunningham (npub10mu…2h9c) for pointing out Lightstone's paper! The other paper is perhaps better for people wanting to think about this issue in lots of ways.
