HW6: Challenge 3 (a,b)
Brittney D'Oleo says: I believe that this could be a possible outline of a proof:
For part (b) I would start with the assumption that f(A) is not bounded. In other words, there has to exist some sub sequences of f(A), f(x_n). such that its limit is infinity. By assumption, we also know that the set A is bounded (now here use Bolzano-W to show that x_n has a convergent sub sequence). Then use the assumption that f is uniformly continuous and so f(x_n) is bounded….but this is a contradiction
Jeremy LeCrone says: Don’t be afraid to use results from other homework problems as well (in particular, recall that one homework problem asks you to prove that if f is uniformly continuous and (x_n) is Cauchy, then (f(x_n)) is Cauchy as well. This observation will help conclude that (f(x_n)) will be bounded…
You must be logged in to post a comment.
https://richmond.qualtrics.com/jfe/form/SV_0kZug50j1agfTff