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### Limits of Sequences

```
Date: 08/19/97 at 08:48:48
From: Suzanne Mooney
Subject: Limits of Sequences

I understand the limit of a sequence <Xsub"n"> to be the number that
the value of the sequence approaches for sufficiently large values of
N, so that way out in the sequence, from then on the terms are
virtually all that number.

But could you please explain to me, preferably in as plain English as
possible, what the limit superior of a sequence <Xsub"n"> is?

Many thanks, Suzanne
```

```
Date: 08/19/97 at 10:21:19
From: Doctor Wilkinson
Subject: Re: Limits of Sequences

If a sequence has an upper bound, then it always has a least upper
bound: this is a fundamental property of the real numbers.

Suppose <Xsub"n"> is a sequence with an upper bound, where n = 1, 2,
3, 4, 5, ...  Then it has a least upper bound: let's call it b1.

Now if you let n start at 2 instead of 1, the resulting sequence still
has an upper bound, since it has fewer terms than the whole sequence,
so it also has a least upper bound: call it b2. b2 may be the same as
b1, or it may be smaller, but it can't be larger. Similarly, you could
start with n = 3, and you'd get a least upper bound of b3.  In this
way, we can define a new sequence b1, b2, b3, ..., and this new
sequence is decreasing.  It may or may not have a lower bound. If it
does have a lower bound, then it has limit, and this limit is the
limit superior.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 08/19/97 at 22:47:59
From: Suzanne Mooney
Subject: Re: Limits of Sequences

Dear Dr. Wilkinson,

Thank you so VERY MUCH for this explanation.  It really gives me a
handle on it, whereas the text definition left me totally in the
dark.

Thanks again,
Respectfully, Suzanne
```

```
Date: 08/20/97 at 10:25:30
From: Doctor Wilkinson
Subject: Re: Limits of Sequences

I'm very glad to have been able to help.  Best wishes.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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