provided that exists
Question. Suppose
and are two sequences of real numbers such that Show that To avoid tedious discussion with infinity, assume in addition that both and are bounded sequences.
Notes. The question is based on the Wikipedia article on limit inferior and limit superior, where the statement above is mentioned explicitly in the article without any proof.
It follows immediately from the definition of
Thus, the question reduces to showing the reversed inequality:
Based on a few basic observations of the definition of liminf, there is one "algebraic" way to show this inequality, which conceals the use of many quantifiers. I will write my own answer below. Alternatives are welcome.
1 answer
We will use the following facts in the proof:
-
(1) If the limit of
exists, so does that of -
(2) If the limit of
as exists, then -
(3) One obvious direction of the inequality is proved by definition:
Proof.
which implies that
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