This is part of a theorem which is left unproved in "Elementary Classical Analysis" by Marsden and Hoffman. Let xn be a sequence in R which is bounded below. Let a be in R. Suppose: (i) For all e > 0 there is an N such that a - e < xn for all n >= N. (ii) For all e > 0 and all M, there is an n > M with xn < a + e. Show that a = lim inf xn. (Definition: When xn is bounded below, lim inf xn is the infimum of the set of all cluster points of xn. If xn has no cluster points, then lim inf xn = + infinity. If xn is not bounded below, then lim inf xn = - infinity.) I was able to use (i) and (ii) to show that a is the limit of a subsequence of xn, hence a is a cluster point. So to show that a = lim inf xn, it is sufficient to show that a is a lower bound for the set of cluster points. This is what I can't do. Any suggestions?
So I set e = a - b in (i) and get b < xn for all n>=N. I don't see the problem with this. I think this implies all cluster points x satisfy b <= x, which means b <= a, but now I'm back where I started.
I think I get it now. Setting e = (a -b)/2 in (i) gives (a + b)/2 < xn for all n >= N ==> (a - b)/2 < xn - b for all n >= N ==> no subsequence of xn can converge to b, since (a - b)/2 > 0 ==> b is not a cluster point. thanks!