Prove that the empty set is a subset of every set.
Prove that the empty set is a subset of every set.
For x an element of R upper k and y an element of R upper k define d1 (x, y) = max{|x of i – y of i| : 1≤i≤k} and d2 (x, y) = min{|x of i – y of i| : 1≤i≤k}. Determine for each of these whether it is a metric or not. Fully explain your answers. Thank you so much.
Give examples to show that the finiteness of the collections in parts c and d is essential. c) for any finite collection G1, G2, ...., Gn of open sets, intersection (at the top of the intersection sign is n and at the bottom is i=1) of Gi is open. d) For any finite collection F1, F2, ...., Fn of closed sets, union sign ( ...continues
Does the closure of a union equal the union of the closures?
See attached file.
Please see the attached file for the fully formatted problems. Let A and B be separated subsets of some Rk, suppose and , and define for . Put , . [Thus if and only if .] (a) Prove that Ao and Bo are separated subsets of R1 . (b) Prove that there exists such that . (c) Prove that ...continues
Give an example in R2 of a convex set. (I know that an example of a convex set is something like a circle or something just as long as you can draw a segment between and 2 pts and the segment still be in the circle, I think, I know something like the packman shape or a peanut is NOT convex) Anyway my teacher wants the answer ...continues
see attached. If f(x) is integrable on E=[0,1] show that...
see attached. Suppose the power series...converges if x=1. Show that the series converges uniformly for...
Hi, I have some questions about the sigma-algebra and looking to having the detail explanation. Thanks alot.
Proving Ruler Function is Riemann Integrable
Define Ruler Function g:(0,1)-> R (it is the usual ruler function available online) as g(x) = { 1/q if x = p/q, p and q are relatively prime { 0 if x is irrational. Show (without using Riemann-Lebesgue Theorem) that g is Riemann integrable. The hint says that since L (lower sum) is zero, it remai ...continues
