Real Analysis Answers

Q: If (bn) is a bounded sequences and lim(an)=0, show that lim(anbn)=0 (explain why the theorem 3.2.3 from book real analysis 3rd edition, by Robert G Bartle can not be used)

Q: Find the limit of the following sequences:
(a) lim2+(1/n2), (b)lim(-1)n/(n+2)
(c)lim((√n-1)/(√n+1))
(d)lim((n+1)/(n(√n)))

Q: Show that the following sequences are not convergent.
(a) (2n), (b)((-1)nn2)

Q. show that if X and Y are sequences such that X converges to x≠ 0 and XY converges, then Y converges.

Q. show that if X and Y are sequences such that X and X+Y are convergent, then Y is convergent.

Q: Give an example of two divergence sequences X and Y such that:
(a) their sum X+Y converges, (b) their product XY converges

Q. For xn given by the following formulas, establish either the convergence or divergence of the sequence X=(xn)
(a) xn= n/(n+1), (b) xn=(〖(-1)〗^n n)/(n+1) ,
(c) xn=n2/(n+1), (d)xn=(2n2+3)/(n2+1)

Q. Prove that set of reals ℝ is an ordered field.

using mathematical induction and find 1^2^3+2^3^4+3^4^5+....upto n terms= n(n+1)(n+2)(n+3)/4

Let A be non empty sebset of real number which is bounded above let -A be the set of all real number -x where x belong A show that sup(A)= -inf(-A)