What separates infinity from the largest number less than infinity?
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The largest number less than another number only has meaning in certain groups of numbers. Infinity isn’t a member of any of those groups, so your question is meaningless.
Consider what the largest number less than 1 would be. On the integers, it’s 0. But what about rational numbers? 0.5 is closer to 1 than 0. 0.9 is closer still. You can get repeatedly closer to 1 by averaging 1 and whatever number you choose less than 1, and you’ll remain in the rational numbers. So the largest number less than x has no meaning in the rational numbers.
It is possible to treat infinity as a cardinal value, similar to the natural numbers (1, 2, 3…), so you may be tempted to ask what infinity – 1 would be. It’s fairly easy to show that infinity – 1 = infinity in the sense that removing one element from an infinite set doesn’t change the number of elements in that set. In order to get back into real numbers, you have to perform an operation like infinity – infinity, which is what’s called an indeterminate form – it can take on any real value, depending on the context.
That might be the answer you’re looking for: there is infinite distance between infinity and any real number, including any you might declare the “largest number less than infinity”.
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