Is there some number that’s larger than .999… but smaller than 1.0?

We can show that the repeating decimal .999… is equal to 1.0 with only a couple of legal moves:

If x = .999…, then 10x = 9.999…

Now subtract x from 10x:

9.999999….

– 0.999999…

You get 9x = 9.0 and thus x = 1

So the repeating decimal  .999… equals 1.0!

Is there some number that’s larger than  .999… but smaller than 1.0?

Such a number would involve an infinitesimal, meaning a literally infinitely small mathematical entity.

— From Everything and More by David Foster Wallace

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