What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333…

So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing.

Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just because the calculator can't do stuff!" but it should at least be proof of comfort for most people.