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Real Analysis
Let x1 > 1 and xn+1 := 2 −
1
xn
for all n ∈ N. Show that {xn} is bounded
and monotone. Find the limit.
Real Analysis
SUPPOSE that f [0,2]-> R is continuous on[02] and differentiableon [0,2] and that f(0)=0 f(1)=1, f(2)=1.... show that there exits c1 belongs to (0,1) such that f (c1)=1
Real Analysis
Determine the local minimum and local maximum values of the function f defined by f(X)=3-5x3+5x4-x5
Real Analysis
State and prove Least Upper Bound (LUB) property of Real Numbers.
Real Analysis
Prove that every non-empty set of real number which is bounded above has the least upper bound (LUB) (supremum)
Real Analysis
Questions
1. Infimum of the set(0,00)
(a) is a non-negative number.
(b) is a positive number.
(c) does not exist.
(d) none of these.
2.Which of the following is not true for a set in R?
(a) A set may not have an infimum in R.
(b) Infimum of a set may not belong to the set.
(c) Infimum and supremum of a set may be equal.
(d) Supremum of a bounded below set always exists in R.
3.Which algebraic property is not true for the set of real numbers R?
(a) For all a
1. Infimum of the set(0,00)
(a) is a non-negative number.
(b) is a positive number.
(c) does not exist.
(d) none of these.
2.Which of the following is not true for a set in R?
(a) A set may not have an infimum in R.
(b) Infimum of a set may not belong to the set.
(c) Infimum and supremum of a set may be equal.
(d) Supremum of a bounded below set always exists in R.
3.Which algebraic property is not true for the set of real numbers R?
(a) For all a
Real Analysis
how to get domain in |x-2|/|x-5|<2?
Real Analysis
Show that 𝑈(−𝑓, 𝑝) = −𝐿(𝑓, 𝑝) and 𝐿(−𝑓, 𝑝) = −𝑈(𝑓, 𝑝)
Real Analysis
Let 𝑓 be differentiable. Show that if lim(𝑥→∞) 𝑓(𝑥) = 𝐿 ∈ ℝ then lim(𝑥→∞) 𝑓 ′ (𝑥) = 0. Provided that the latter limit is existing. Give an example where the converse is not true. Also give an example for which the limit of 𝑓 ′ is not existing even though the limit of 𝑓 is the same as given
Real Analysis
Which of the following sets Sj , (j = 1, . . . , 4) are NOT a neighbor-
hood of the given point a.
(1) S1 = (−2, 3), a = 1; (2) S2 = [−3, ∞), a = 2.
(3) S3 = {x: |x − 1| ≤ 2} , a = 3; (4) S4 = {x: |x − 3| ≥ 2} , a = 1.
hood of the given point a.
(1) S1 = (−2, 3), a = 1; (2) S2 = [−3, ∞), a = 2.
(3) S3 = {x: |x − 1| ≤ 2} , a = 3; (4) S4 = {x: |x − 3| ≥ 2} , a = 1.
