Algebra Worksheet

1. Let R+ denote the set of positive real numbers.(a) Show that the continuous function f : R+ -+ R given by /(x) =1/(l+x) is bounded but has neither a maximum value nor a minimumvalue2. Let X denote t he subset (-1, 1) x 0 of R2, and let U be the open ballB(O; 1) in R2, which contains X. Show there is no E > 0 such that theE-neighborhood of X in Rn is contained in U. Math 127C Homework 2
1.
(6 points) Show that a finite union of compact sets is compact. Give an example of an
infinite union of compact sets which is not compact.
2.
(6 points) Let f : [0, 1] → R3 be a continuous function. Let R3 have the usual Euclidean
coordinates (x, y, z), and suppose f (0) = (−10, 5, 7) and f (1) = (10, 3, 3). Show that Im(f )
intersects the plane in R3 given by x = 0.
3.
(6 points) Munkres 1.4.1(a)
4.
(7 points) Munkres 1.4.2