Define the set of all non-zero real number as $A = \{a \in \mathbb{R} : a \neq 0\}$. So the obvious gap of the set is the point zero. I am a bit confused by the definition of upper/lower bound, because a tutor of my class said this set $A$ is bounded. Does this count as a lower bound or upper bound?
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2This set is unbounded both above and below. That's clear from the definition. Please check that this is exactly what your tutor told you. Perhaps what was meant is the $0$ is a lower bound for the set of all positive numbers. – Ethan Bolker Feb 03 '22 at 15:20
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To emphasize... if we were to assume that the set was bounded, then that would mean that there would need to be some positive number $M$ such that $|a|<M$ for all $a\in A$... but $M\in A$ as well and clearly $|M|<M$ is false showing that no such bound exists that works for all $a\in A$. – JMoravitz Feb 03 '22 at 15:47