Show that if M1 and M2 are least upper bounds for the sequence {an}, then M1 = M2. That is, a sequence cannot have two different least upper bounds.

If sequence a_n of positive numbers in increasing, then it is not bounded. TRUE or FALSE

Find the first four terms of each of the recursively defined sequences(Discrete math)

Exercise 2 Use the formal definition of a Cauchy sequence to show that the sequence (an) sequence. +31 = 4n+1 √n=1 is a Cauchy

Give a recurrence relation that would fit the sequence a1, az, A3, A4, A5, .= 1,2, 5, 26, 677 ...

Show that a sequence (an) is bounded if and only if it is bounded above and below.

Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers. For (i) take = xn= Show that {xn} is a Cauchy but not monotonic. For(ii), take= xn=n Show that{xn} is monotonic. Take &=;n E N; and n > N and m=n+1 Explain why (-1)" n |xn−Xn+1| > E Conclude that {xn) is monotonic but it is not Cauchy. For (iii), take xn = 1 +(-1)^. Show that {xn} is bounded, i.e. show that |xn| <2; for all n E N. Compute |xn-Xn+1| Explain why {xn} cannot be a Cauchy sequence. For (iv) Recall that every Cauchy sequence is bounded. Conclude that no such an example of a sequence {x} can be constructed.

4. Suppose that (an) is a sequence with an > 0 for all n. Define a sequence (bn) by + An a₁ + a₂ + n Show that bn diverges. bn =