By Derek G. Ball and C. Plumpton (Auth.)

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**Sample text**

An} is a sequence with lim an = oo. If {bn} is the sequence defined by bn = -αΛ for all n, show that lim bn = - oo. 8. Give examples of convergent sequences for which: (i) the limit is different from all the terms of the sequence; (ii) the limit is equal to all the terms of the sequence; (iii) the limit is equal to an infinite number of terms of the sequence, but not all of them; (iv) the limit is equal to a finite number of terms of the sequence. 9. {an} and {bn} are sequences and lim an = l, lim bn = oo.

Therefore for all η^ Ν, l-o Thus 1 jn -* 0 as n n 1 ^ JL^M 44 AN INTRODUCTION TO REAL ANALYSIS (ii) Here every term of the sequence is equal to the number c. Given ε > 0, we may choose N to be any natural number we please. Then, for all n^ N9 \C—C\ = 0 < 8. (iii) If 0 < |JC| < 1, then | * | = l/(l+j>) for some y > 0. 3 there is a rational MfN satisfying 0 < M/N < ey. Thus 0 < — =^ —: < εv and so t-r- < ε. J N N Ny So for all η^ Ν, \x*-0\ = \x*\ = \ x \ ^ ± ^ ± ^ e . Therefore x" -- 0 as « ->- ~ .

6. Give an example of a set of real numbers which is bounded but for which neither the least upper bound nor the greatest lower bound belong to the set. 7. Show that there is no greatest rational whose square is less than 2, and no least positive rational whose square is greater than 2. The set A consists of all rationals whose square is less than 2. Show that A is bounded but that there is no least rational upper bound. What is the least upper bound for A ? What is the greatest lower bound for A ?