let {xn}n tends to infinity and {yn}n tends to infinity be two convergent (in R) sequences such that xn>or=yn for all natural numbers n. Show that:
as both tend to infinity lim xn>or=lim yn
I'm doing some review and came across this question, I have on which is similar except instead of limits it's for the supremum of the sequences. Would the proofs be similar?
Alright, I tried it but I could not seem to get anywhere. Also the proof I had done for the supremums was in extended reals, not just reals, although that just rules out infinity. I'm not sure why but I'm just having a hard time with this. Can you provide some more guidance?
This is very basic stuff though annoying to write it and show it , but let's see:
== First, if is a convergent sequences of non-negative terms then , otherwise: choose (note that this number is positive), then , which contradicts the data that is a non-negative sequence
== Second...we're done, since is a non-negative sequence converging (by arithmetic of limits) to ...
Tonio