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Limit Superior and Limit Inferior of a Bounded Real Sequence

Every undergraduate student who has done a basic course on real analysis learns something about limit points of a real sequence. We know that a bounded real sequence has at least one limit point. And we also know that the set of all limit points of a sequence (which is obviously bounded) has its supremum and infimum within itself. That is to say, the supremum and infimum of the set of all limit points of the bounded sequence are themselves limit points of the sequence. The supremum is called "limit supremum" and the infimum is called "limit infimum" of the sequence. This is how many textbooks on analysis define the two terms. However other textbooks define them as:

\[\lim\sup \{x_n\} = \inf_{n \ge 0} \sup_{m \ge n} \{x_m\} = \inf \{ \sup \{ x_m : m\ge n\} : n\ge 0\}\]

\[\lim\inf \{x_n\} = \sup_{n \ge 0} \inf_{m \ge n} \{x_m\} = \sup \{ \inf \{ x_m : m\ge n\} : n\ge 0\}\]

Try to establish an equivalence between these two definitions. When I learnt these, I failed to find such an attempt to establish the aforesaid equivalence even in good texts on the subject. The proof, as I had to chalk myself out at that time, was not too difficult, but was a cumbersome one and required considerable mental gymnastics. So, give it a try. I will upload the answer if I don't find a satisfactory one coming up within a week.

Note by Kuldeep Guha Mazumder
2 years ago

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