PropositionIf a sequence is

eventuallybounded, then it is bounded.

Proof

Suppose $\displaystyle a_n \rightarrow a$ as $\displaystyle n \rightarrow \infty$. Then $\displaystyle \forall \epsilon > 0\ \exists N(\epsilon) \in \mathbb{N}\ s.t.\ |a_n - a| < \epsilon \ \forall n > N$. That is, the subsequence of $\displaystyle a_n$ defined by $\displaystyle a_{N + n}$ is bounded ($\displaystyle |a_{N + n} - a| < \epsilon$). Since $\displaystyle a_{N + n}$ is bounded $\displaystyle \forall n>N$ it follows that $\displaystyle a_n$ is eventually bounded.

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From here I have no idea where to go. If I'm right, I need to show that as a result of this sequence being eventually bounded, it must be consequently bounded. But I don't see how the first statement implies the second. It says in the margin that I should consider the idea that every set has a maximum and minimum element but at the moment I'm not sure on anything really. Any advice is welcome And if this is all completely wrong, please inform me about it.

Also I should note that I have just started my first year at university so a lot of things are a bit alien to me at the moment. Please be patient if I don't understand something that may be above my level