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2. ** Prove or give a counterexample (a) If AC R is nonempty and open then...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn C is open. b) Give an example to show that the intersection of infinitely many open sets may not be open.
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y...
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y of R such that (Y, dy) contains nonempty closed and bounded subsets which are not compact (here dy is the metric inherited from the usual metric in R)
Using only the definition of compact sets in a metric space, give examples...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
1. Let A be a nonempty subset of R such that every number in A is...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
Give an example of infinitely many sets of real numbers, called such that all four conditions...
Give an example of infinitely many sets of real numbers, called
such that all four conditions are satisfied at once. They are: i)
each set is bounded above. ii)
for all m and all n. iii) the intersection of and is empty whenever
m and n are not equal. iiii) for all n, is not an
element of .
Not sure what to do here, but I believe it can be done using the
fact that there is infinitely many...
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Let r be any rational number and define L = { x in Q: x <...
Let r be any rational number and define L = { x in Q: x < r }, the set of rational numbers less than r. Show that L is a Dedekind cut by proving the following properties: A. There exists a rational number x in L and there exists a rational number y not in L. ( This proves L is nonempty and L is not equal to Q) B. If x in L, then there exists z in...
1. Let a,b ER with a < b. In this problem we are going to prove...
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
5- Recall that a set KCR is said to be compact if every open cover for...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at le...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at least punctually in its domain? b)If {fn (2.)) is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Is {fn (x)) a succes- sion of functions uniformly bounded?
For the following statements give a counterexample or demonstrate them: a)If fn (x) is...
Question 6. a) Let A, B and C be open subsets of R. Prove that AnBn C is open. b) Give an example to show that the intersection of infinitely many open sets may not be open.
Using only the definition of compact sets in a metric space, give examples of: (a) A nonempty bounded set in (R", dp), for n > 2 and 1 < pく00, which is not compact. (b) A bounded subset Y of R such that (Y, dy) contains nonempty closed and bounded subsets which are not compact (here dy is the metric inherited from the usual metric in R)
Using only the definition of compact sets in a metric space, give examples...
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
1. Let A be a nonempty subset of R such that every number in A
is greater than 2 (NOTE: This doesn’t necessarily mean that A =
(2,∞)).
(a) Explain why A must have an infimum.
(b) Let c = inf(A). Prove that a∈A INTERSECTION (−∞,a] =
(−∞,c].
CAN SOMEONE PLZ HELP ME WITH THIS QUESTION.
1. Let A be a nonempty subset of R such that every number in A is greater than 2 (NOTE: This doesn't necessarily mean...
Give an example of infinitely many sets of real numbers, called
such that all four conditions are satisfied at once. They are: i)
each set is bounded above. ii)
for all m and all n. iii) the intersection of and is empty whenever
m and n are not equal. iiii) for all n, is not an
element of .
Not sure what to do here, but I believe it can be done using the
fact that there is infinitely many...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
1. Let a,b ER with a < b. In this problem we are going to prove that the open interval (a, b) containes infinitely many rational numbers by following these steps (a) First let NEN be an arbitrary rational number. Use the density of the rational numbers to show that (a, b) contains N rational numbers. There is a hint about this in the lecture on the density of rationals.) (b) Now uppose that there are finitely many rational numbers...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at least punctually in its domain? b)If {fn (2.)) is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Is {fn (x)) a succes- sion of functions uniformly bounded?
For the following statements give a counterexample or demonstrate them: a)If fn (x) is...
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