Suppose A CR is countable. Show that there exists a real number I such that An...
Complex Analysis: Suppose f(2) is an entire function and that there exists a real number Ro such that \f(2)] = 2l for any complex number z with [2] > Ro Prove that f is of the form f(x) = a + bz for all z E C.
2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number . (ii) Show that sx -2Nan for N E N is a Cauchy sequence 2. Suppose that (an), İs a sequence of complex numbers such that there exists a positive number 0 such that for all NEN an M (i) Show that (ON)N converges to a number...
1. Show that if A and B are countable sets, then AUB is countable. 2. Show that if An are finite sets indexed by positive integers, then Un An is countable. 3. Show that if A and B are countable sets, then A x B is countable. 4. Show that any open set in R is a countable union of open intervals. 5. Show that any function on R can have at most countable many local maximals. Us
proofs For this assignment, know that: An integer is any countable number. Examples are: -3, 0, 5, 1337, etc. A rational number is any number that can be written in the form a/b, a and b are integers in lowest terms, and b cannot equal 0. Examples are 27, 22/7, -3921/2, etc. A real number is any number that is not imaginary or infinity. Examples are 0, 4/3, square root of 2, pi, etc. 1. Prove or disprove: There exists...
13. An algebraic number is a real number which is the root of a polynomial co + ciz c2n in which all of the coefficients c i 1,2,.,n) are integers. The order of an algebraic number is the smallest natural number n for which z is a root of an n-th degree polynomial with integer coefficients. A real number is transcendental if it is not algebraic. a) Show that the set of algebraic numbers of order n is countable (b)...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
Suppose fon (0,1) is uniformly continuous. Show that there is a real number A such that the function F defined by F(O)=A, F(x)=f(x) if x € (0,1), is continuous on (0,1]. (Suggestion: Show first that if {Xn}, Xn € (0,1] has lim xn = 0, then {f(xn)} is a Cauchy no sequence. Then show this sequence has the same limit no matter which {Xn} sequence going to you choose).
Identify the correct steps involved in proving that the union of a countable number of countable sets is countable. (Check all that apply.) Check All That Apply Since empty sets do not contribute any elements to unions, we can assume that none of the sets in our given countable collection of countable sets is an empty set. If there are no sets in the collection, then the union is empty and therefore countable, Otherwise let the countable sets be As,...
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a) 5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...