Example:
If 
is the sequence of all
rational numbers in the interval [0, 1], enumerated in any way, find the lim sup and
lim inf of that sequence.
Since the numbers 1 and 0 are itself rational numbers, it is clear that
- sup{ an } = 1 and
inf{ an } = 0
Therefore, we already know that
-
0
lim inf{ an }
lim sup{ an }
1
To find the lim sup, we will construct a subsequence that
converges to 1:
- there exists
0 < aj1 < 1 with
1 - aj1 < 1
- there exists
0 < aj2 < 1 with
1 - aj2 < 1/2 and
aj1 # aj2
- there exists
0 < aj3 < 1 with
1 - aj3 < 1/3 and
aj3 different from the previous ones
- and so on ...
These numbers exist because the rational numbers in the interval
[0, 1] are arbitrarily close to any real number in that interval,
according to the Density principle.
The subsequence { ajk } constructed
in the above way converges to 1. We already know that any limit
of any convergent subsequence must be less than or equal to 1.
Therefore, since the lim sup is the greatest limit of any
convergent subsequence, we have
Similarly, we can extract a subsequence
{ ajk } that converges to 0. We
also know that every limit of any convergent subsequence must be
greater or equal to zero. Therefore, since the lim inf is the
smallest possible limit of all convergent subsequence, we
have:
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