let xn be a sequence such that there exists a 0 c 1 such that

(n2?1 n2. ) is Cauchy using directly the definition of. Cauchy sequences. Let {xn} be a sequence such that there exists a 0 <C< 1 such that.

A sequence (an)n=12

Show that X is not a bounded sequence and hence is not convergent. Solution. Since lim. (xn+1 xn. ) = L given ? > 0

Suppose xn is a bounded sequence in R. ?M such that but there are no such pt. ... Let V = C([01])=all continuous functions on the interval.

Suppose that (xn) is any sequence in A with xn = c that converges to c and let ? > 0 be given. From. Definition 6.1

Proposition 3.19. A convergent sequence is bounded. Proof. Let (xn) be a convergent sequence with limit x. There exists N ? N such that.

(b) Let E ? R be a subset such that there exists a sequence {xn} in E with Solution: For any x = 0 there exists an N such that

converges to 0 then the sequence (xn n) must converge to 0. Solution: The given statement is TRUE. If xn ? 0

continuous at c if for every ? > 0 there exists a ? > 0 such that In particular f is discontinuous at c ? A if there is sequence (xn) in the domain.

Let us now state the formal definition of convergence. Definition : We say that a sequence (xn) converges if there exists x0 ? IR such that for every.

(c) If {xn} is a sequence of real (or complex) numbers that converges to Now suppose {xn} converges to x i e for all ? > 0 there exists N ? N such 

Proposition 3 19 A convergent sequence is bounded Proof Let (xn) be a convergent sequence with limit x There exists N ? N such

xn = s Proof Let ? > 0 be given Since limn?? an = s there exists a positive integer N1 such that

Suppose xn is a bounded sequence in R ?M such that but there are no such pt Let V = C([01])=all continuous functions on the interval

converges to 0 then the sequence (xn n) must converge to 0 Solution: The given statement is TRUE If xn ? 0 then there exists n0 ? N such that xn < 1

Let us now state the formal definition of convergence Definition : We say that a sequence (xn) converges if there exists x0 ? IR such that for every

(b) Let E ? R be a subset such that there exists a sequence {xn} in E with the property that xn ? x0 /? E Show that there is an unbounded continuous 

Let X = (xn) be a sequence of positive real numbers such that lim (xn+1 xn ) Let (fn) ? C[01] be such that there exists M > 0 such that fn ? ?

Let A be a bounded subset of R Show that there exists a sequence (an) of elements of A such that lim(an) = sup(A)

Let {an} be a bounded sequence such that every convergent subsequence of {an} has a limit L Prove that limn?? an = L Solution Method 1: Note that La = {L}