Show sequence is bounded in an interval

[darkestar]

Homework Statement

Show that a sequence {a_n} is bounded if and only if there is an interval [c, d] such that {a_n} is a sequence in [c,d].

Homework Equations

A sequence {a_n} is bounded provided that there is a number M such that |a_n| <= M.

The Attempt at a Solution

http://img180.imageshack.us/img180/7856/001hep.th.jpg

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[jgens]

The first part looks fine. I'm having difficulty reading the second part, but it looks like you assume the c=-d, which isn't necessarily a valid assumption.

 

[darkestar]

For the second part I let c = -d because I was trying to end with a result such as the sequence being in the interval [-M, M]. If I leave it as the interval [c, d] then I'm not sure how to end with the conclusion that every element of the sequence is less than a number M.

 

[jgens]

Consider max{|c|,|d|}.

 

[darkestar]

So I can define M to be max{ |c|, |d| }.

Therefore, |a_n| <= M.

Would this be all that I need to write for the proof in that direction?

 

[jgens]

It might be nice if you write things out a little more explicitly, but yes, it's all that you need.

 

[darkestar]

I'm not really sure how I should be more explicit for the proof.

 

[jgens]

-\max\{|c|,|d|\} \leq c \leq a_n \leq d \leq \max\{|c|,|d|\}