Eventually bounded/bounded sequences - I can't complete the proof -_-

**Proposition**

If a sequence is *eventually* bounded, 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 :)

Re: Eventually bounded/bounded sequences - I can't complete the proof -_-

Your initial proof attempt here begins with the assumption that $\displaystyle a_n$ converges. That's not justified by the assumption that $\displaystyle a_n$ is eventually bounded. For instance, $\displaystyle a_n = (-1)^n$ is certainly eventually bounded, but it doesn't converge.

Try to state what "eventually bounded" means.

Re: Eventually bounded/bounded sequences - I can't complete the proof -_-

Quote:

Originally Posted by

**Femto** **Proposition **If a sequence is *eventually* bounded, then it is bounded.

**Proof** Suppose $\displaystyle a_n \rightarrow a$ as $\displaystyle n \rightarrow \infty$.

Bounded sequences may not converge.

But from the given $\displaystyle \exists B>0~\&~\exists N\in\mathbb{Z}^+$ such that if $\displaystyle n\ge N$ then $\displaystyle \left|a_n\right|\le B$.

Let $\displaystyle B' = \sum\limits_{k = 1}^{N - 1} {\left| {a_k } \right|} + B$ do we have a bound for $\displaystyle a_n~?$

Re: Eventually bounded/bounded sequences - I can't complete the proof -_-

Quote:

Originally Posted by

**Plato** Bounded sequences may not converge.

But from the given $\displaystyle \exists B>0~\&~\exists N\in\mathbb{Z}^+$ such that if $\displaystyle n\ge N$ then $\displaystyle \left|a_n\right|\le B$.

Let $\displaystyle B' = \sum\limits_{k = 1}^{N - 1} {\left| {a_k } \right|} + B$ do we have a bound for $\displaystyle a_n~?$

Thanks.

Yes I realise that bounded sequences may not converge but I thought that all I have to prove is that a sequence is bounded because it is eventually bounded, so I could consider a case in which a sequence is eventually bounded. But yes I see what you mean - that wouldn't be correct.

I've defined $\displaystyle B = max(|a_1|, |a_2|,...,|a_N|, M)$ as my bound for $\displaystyle a_n$. I have seen what you've written before but, and do forgive me if this is a stupid question, what is $\displaystyle B'$ exactly? I see you've defined $\displaystyle B$ to be the bound that exists for all $\displaystyle n \ge N$ but why is $\displaystyle B'$ represented as a sum of bounds? If it is the bound for the whole sequence then shouldn't it be the maximum taken from a set of terms rather than a sum (sorry it just confuses me)?

Re: Eventually bounded/bounded sequences - I can't complete the proof -_-

Quote:

Originally Posted by

**Femto** I've defined $\displaystyle B = max(|a_1|, |a_2|,...,|a_N|, M)$ as my bound for $\displaystyle a_n$. I have seen what you've written before but, and do forgive me if this is a stupid question, what is $\displaystyle B'$ exactly? I see you've defined $\displaystyle B$ to be the bound that exists for all $\displaystyle n \ge N$ but why is $\displaystyle B'$ represented as a sum of bounds? If it is the bound for the whole sequence then shouldn't it be the maximum taken from a set of terms rather than a sum (sorry it just confuses me)?

Both ways work. I just always use the way I did it.

$\displaystyle \forall n$ it is true that $\displaystyle |a_n|\le B'$.