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?