What is the definition of a sequence that is not bounded below? I know that if a sequence is bounded below if there exists a number $m$ such that $m \leq a_n$ for every $n$. But to say it's not bounded below would you say, if for every $m$ such that $a_n < m$ for all $n$? But isn't that just like the definition of an upper bound?
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$\begingroup$You have the equivalent statment just slightly wrong, and it is causing your confusion.
By the definition, a sequence $a_n$ is not bounded below if there is no $m$ such that {$m\leq a_n$ for every $n$}. I have added those { } to try to make the meaning more unambiguous.
The contrapositive of that would be that
"For every $m$, there exists some $n$ such that $a_n < m$."
Compare that to the definition of an upper bound $m$ which is:
"For every $n$, $a_n \leq m$."
The difference is more than the distinction between $<$ and $\leq$, in that the the second statement is a statement about a property of all possible $n$, while the first talks about the existence of some special $n$ with the stated property.
$\endgroup$ $\begingroup$Okay now that you know the definition of "Bounded" (there exists an $M$ such that $M\le a_n$, $\forall a_n \in A$. So let's try to reverse this. This goes back to if $(\exists P)^\neg \cong \forall P$, likewise $(\forall Q)^\neg \cong \exists Q$. So if a sequence is unbounded then there exists an $a_n\in A$ such that for every $M$ we have $a_n < M$.
Notice how the existence and for all have reversed. While this is similar to bounded above (there exists an $M$ such that $M\ge a_n$, $\forall a_n \in A$), it isn't the same.
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