Deciding Bounds & Finding Sup/Inf

[elle]

Help! I've been asked to decide whether or not the given sequence is

i) bounded above
ii) bounded below
iii) bounded

and to determine (where appropriate) the supremum and infimum.

I've got like 6 similar questions to attempt but I don't know where to start The examples we have been given in our notes are so complicated that I don't understand them Can someone show me how to approach the answer? Thanks very much!

I know Supremum is the least upper bound and infimum is the greatest lower bound.

http://tinypic.com/ftlrg5.jpg"

 

[Benny]

Well quite clearly, since n is a natural number we have a_n = 1 + \frac{1}{n} \ge 1 + \frac{1}{{n + 1}} = a_{n + 1}. So the 'non-alternating' part is a decreasing sequence. In fact, that part of your sequence will converge to 1. The behaviour of (-1)^n for natural numbers n is fairly obvious. You should now be able to decide on a bound.

 

[HallsofIvy]

First make certain that you know what "bounded above", "bounded below", and "bounded" MEAN!

Then write out a few terms of the sequence. It should quickly become obvious.