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_introanalysis_assignment_1.pdf

_introanalysis_assignment_2.pdf

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en
t-1
Introduction to Analysis — IAUSS 2020
Assignment-1
Due: TBD
Marks
[12]
(a) If a and b are any two real numbers, then prove that
|a − b| ≥ ||a| − |b|| and |a + b| ≥ ||a| − |b||
(b) If a nonempty set S of real numbers is bounded below, then infS is the unique real number
α such that
m
i. x ≥ α for all x in S
ii. if > 0 (no matter how small), there is an x0 in S such that x0 < α + (c) Find the supremum and infimum of each S. State whether they are in S [5] ig n i. S = {x|x = −(1/n) + [1 + (−1)n ]n2 , n ≥ 1} ii. S = {x|x2 ≤ 7} iii. S = {x||2x + 1| < 5} 2. Let n0 be any integer (positive, negative, or zero). Let Pn0 , Pn0 +1 , · · · , Pn , · · · be propositions, one for each integer n ≥ n0 , such that (a) Pn0 is true (b) for each integer n ≥ n0 , Pn implies Pn+1 As s Then prove that Pn is true for every integer n ≥ n0 . [7] 3. (a) Find Find and prove by induction an explicit formula for an if a1 = 1 and for n ≥ 1 an i. an+1 = (n + 1)(2n + 1) n 1 ii. an+1 = 1 + an n (b) Prove by induction that sin(x) + sin(3x) + · · · + sin(2n − 1)x = [6] 1 − cos(2nx) , n≥1 2 sin(x) 4. (a) Prove that a set T1 is denumeratble if and only if there is a bijection from T1 onto a denumerable set T2 . –2– (b) Prove in detail that if S and T are denumerable, then S ∪ T is denumerable. (c) Prove that the collection F (N) of all finite subsets of N is countable. [30] T. Sheel, August 7, 2018 Introduction to Analysis — IAUSS 2020 Assignment-2 Due: TBD en t-2 Marks 1. Let A and B be bounded nonempty subsets of R, and let A + B = {a + b : a ∈ A, b ∈ B}. Prove that sup(A + B) = supA + supB and inf (A + B) = inf A + inf B. [8] 2. (a) Use the definition of the limit of a sequence to establish the following limits n =0 i. lim n→∞ n2 + 1 2 n −1 1 ii. lim = 2 n→∞ 2n + 3 2 √ n iii. lim =0 n→∞ n+1 (−1)n n iv. lim =0 n→∞ n2 + 1 n+1 a + bn+1 3. (a) If 0 < a < b, determine lim n→∞ an + bn a+b p (n + a)(n + b) − n = (b) If a > 0, b > 0, show that lim
n→∞
2
(c) Show that if (xn ) and (yn ) are convergent sequences, then the sequences (un ) and (vn )
defined by un = max{xn , yn } and vn = min{xn , yn } are also convergent.
4. (a) Let x1 > 1 and xn+1 = 2 −
As
s
[5]
ig
n
[9]
m
[4]
the limit
1
for n ∈ N. Show that (xn ) is bounded and monotone. Find
xn
(b) Let A be an infinite subset of R that is bounded above and let u = supA. Show there
exists an increasing sequence (xn ) with xn ∈ A for all n ∈ N such that u = lim (xn ).
[4]
n→∞
5. Let X = (xn ) and Y = (yn ) be given sequences, and let the ”shuffled” sequence Z = (zn ) be
defined by z1 = x1 , z2 = y1 , · · · , z2n−1 = xn , x2n = yn , · · · . Show that Z is convergent if and
only if both X and Y are convergent and lim X = lim Y .
[30]
T. Sheel, August 7, 2018