I have proven that if {a_{n}} and {b_{n}} are 2 sequences such that a_{n} >= b_{n} for each n>n_{0} , for some n_{0} belongs to positive integers with limit of an when n goes to infinity is L and limit of bn when n goes to infinity is M , then L >= M by using a contradiction proof.
My problem is I think that we can say, if a_{n} > b_{n} for each n > n_{0} ,for some n_{0} belongs to positive integers, then L > M. But I'm having a difficulty in proving that claim. I have tried to prove it in the same way by using the contradiction proof but I get stuck because then I assume L <= M and as in earlier proof when I get (M-L)/2 as epsilon I get an epsilon which is >= 0 but epsilon should be strictly greater than 0.
So, is there another way of proving this? Please can somebody help?
In general, the best you can say for limits is or . That is, if (strictly less than) for all , then, using the indirect proof that Plato suggested, we can show that (less than or equal to).