By Krantz S.G.

ISBN-10: 148222075X

ISBN-13: 9781482220759

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**Extra info for Foundations of analysis**

**Sample text**

If {aj } is any subsequence of the given sequence then α ≤ lim inf aj ≤ lim sup aj ≤ β . 3. LIM SUP AND LIM INF 33 and another sequence {ank } such that lim ank = β . k→∞ Proof: For simplicity in this proof we assume that the lim sup and lim inf are finite. We begin by considering the lim inf. There is a j1 ≥ 1 such that |A1 − aj1 | < 2−1 . We choose j1 to be as small as possible. Next, we choose j2 , necessarily greater than j1 , such that j2 is as small as possible and |aj2 −A2 | < 2−2 . Continuing in this fashion, we select jk > jk−1 such that |ajk −Ak | < 2−k , etc.

Let > 0. Then there is an integer N > 0 such that for j > N we have the inequality |aj − α| < /2. Likewise, there is an integer N > 0 such that for j > N we have |aj − α| < /2. Let N0 = max{N, N}. Then, for j > N0 , we have |α − α| ≤ |α − aj | + |aj − α| < /2 + /2 = . Since this inequality holds for any > 0 we have that α = α. So the limit of the sequence is unique. Next, with α the limit of the sequence and = 1, we choose an integer N > 0 such that j > N implies that |aj − α| < = 1. For such j we have that |aj | ≤ |aj − α| + |α| < 1 + |α| ≡ P .

What does it mean when the lim sup and lim inf are equal? Exercises 1. Consider {aj } both as a sequence and as a set. How are the lim sup and the sup related? How are the lim inf and the inf related? Give examples. 2. Prove the last proposition in this section. 3. How are the lim sup and lim inf of {aj } related to the lim sup and lim inf of {−aj }? 4. Let {aj } be a real sequence. Prove that if lim inf aj = lim sup aj then the sequence {aj } converges. Prove the converse as well. 5. Let a < b be real numbers.

### Foundations of analysis by Krantz S.G.

by Richard

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