I have played around with this question for my analysis class and I would like some help:
Given: L= lim f1(x), M=lim f2(x)
Show that if f1<=f2 for all x in some interval (a,b), then L<=M
It's probably easiest to do this by contradiction. Suppose that L>M and let . For x sufficiently close to some limit point c (the limit point isn't specified in the question), it will be true that and . From that, and the fact that , you should be able to use the triangle inequality to get a contradiction.