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I cannot conclusively answer this as I have two different opinions:

  1. It is false considering that the set of integers is a subset of the reals and that the “smallest” integer is also a real, hence there is a real number which does not have a smaller integer
  2. It is true, considering that there are infinite integers, so there is always an integer smaller than a real

Could someone explain this to me?

Asaf Karagila
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J. Cricks
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    argument $1$ is false. It is not true that every subset of the reals has a least element. the integers, in particular, have no least element. – lulu Mar 23 '20 at 11:51
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    How would u prove it? – J. Cricks Mar 23 '20 at 11:55
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    Should have said: your argument $2$ is also false, though the conclusion is correct. To see that $2$ is false note that there are infinitely many positive integers but there is no positive integer smaller than $0$, say. – lulu Mar 23 '20 at 11:59
  • If all you are looking for is intuition... then pick a real number, any real number. After I've heard your number and am given a brief moment to think, I'll be able to say an integer smaller than what you just said. Let's try a few times... If you say $5.8$ I might say $5$. If you say $-\pi$ I might say $-4$. If you say $-10000000000000.37$, I might say $-10000000000001$. Do you need more examples to see what I'm doing? Can you see that regardless what you say for your real number I can always pick an integer less than it? – JMoravitz Mar 23 '20 at 12:22
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    As for "how would u prove it?" if you are talking about how the integers has no least element... suppose for contradictory purposes that it did have a least element, $\ell$. Then, consider $\ell-1$ which we understand will also be an integer and in particular will be smaller than $\ell$, contradicting the notion that $\ell$ were in fact least. – JMoravitz Mar 23 '20 at 12:25
  • Does "smaller than" mean "less than", or are we talking about magnitude, so that $0$ is smaller than either $8$ or $-3$? Assuming the first, the link provided by lulu is a good one. Assuming the second, $0$ ends up being "smaller" than any real number (except for $0$, of course). – John Hughes Mar 23 '20 at 12:26

1 Answers1

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Let $x \in \mathbb{R}$.

If $x \in \mathbb{R}$ but $x \notin\mathbb{Z}, [x]$ is an integer smaller than $x.$

If $x \in \mathbb{Z}, x-1$ is an integer smaller than $x.$

SARTHAK GUPTA
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