# Category Archives: Challenges

## HW 9: Challenge 2(a,b,c)

Tongzhou, 11/21 (a) Since needed to be determined firstly. By definition, . Since is any partition of , Therefore, . (b) Sidework: I want to have which means I want to determine how far could be away from . Thus so that . Now choose . Again I need to determine before I find . […]

## HW 8: Challenge 3(a,b)

Consider the sequence of functions defined by a) Prove that converges uniformly on and find . In order to better examine this, look at , then . The limit of as is 0. This holds similarly for For and , so converges pointwise to Let be arbitrary and choose (which exists by Archimedean Property). Let both be […]

## HW 7: Challenge 3

Problem: Let be a differentiable function on an interval . Show that if  on , then  is one-to-one. Provide an example to show that the converse of the statement need not be true. Proof:  Suppose  is a differentiable function on some interval . Let for some arbitrary . By Theorem 5.2.3, is also continuous on and so […]