This is part of a theorem which is left unproved in "Elementary Classical Analysis" by Marsden and Hoffman.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Sufficient conditions for a = lim inf xn

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