Real Analysis Lecture 12
Maksim Maydanskiy
Fall 2012
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Properties of lim inf and lim sup.
Last time for a bounded sequence sn we defined vN = sup{sn |n > N } and
uN = inf{sn |n > N } and since vN and un are bounded monotone, they converge,
so we can define lim sup sn = lim vN and lim inf sn = lim uN .
Note that vN ≥ uN , so lim sup sn ≥ lim inf sn .
We can extend these definitions to the case when sn is not bounded. For example, when sn is not bounded above, we have vn = +∞ and we put lim sup sn =
+∞. Similarly, for a sequence not bounded below, uN = −∞ and we put
lim inf sn = −∞. Not that the converse also holds - if lim sup sn = ∞ then
sn is not bounded from above, and if lim inf sn = −∞ then sn is not bounded
from below.
There is however a another issue.
Example 8: sn = n This is not bounded above, so lim sup sn = ∞. But note
uN = N + 1, which also diverges to −∞. So lim inf sn = −∞.
Example 9: sn = n if n is odd and − n1 if n is even. Then lim sup sn = ∞ and
lim inf sn = 0 (check this!).
So in general lim sup sn and lim inf sn always make sense, and lim sup sn ≥
lim inf sn but either or both may be infinite. We shall concentrate on bounded
sequences, making only occasional remark about the unbounded ones.
To get a bit more intuition about lim inf and lim sup, we have the following
propositions:
Proposition 1.1. If B > s = lim sup sn then there exists N such that sn < B for
all N > n.
Proof. As vN are decreasing converging to s we get vN < B for some N (Details:
For large N we have |vN − s| = vN − s = ε < B − s, so vN < B). Then as vN
is a supremum, sn ≤ vN < B for all n > N .
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Proposition 1.2. If B < lim sup sn then for any N there is n > N with sn > B.
That is, sn > N infinitely often.
Proof. As vN decrease to a value above B, we get vN > B for all N . Since vN is
the supremum of {sn |n > N }, it is the lowest upper bound, so B > vN is not an
upper bound, which says we can get sn above B for some n > N . This holds for
arbitrary N so we are done.
So lim sup sn is the divider between those B for which sn is eventually below
them and those for which sn goes above B infinitely often.
Note that for lim sup itself it can go either way - the sequence may never go
above it, like for the sn = 1 − n1 or always go above it like sn = n1 , or alternate as
sn = (−1)n n1 , or equal it all the time like the constant sequence, or equal some of
the time and not equal another part - anything goes really.
We have similar propositions for lim inf sn .
Proposition 1.3. If b < i = lim inf sn then there exists N such that sn > i for all
N > n.
Proposition 1.4. If b > i = lim inf sn then there are infinitely many sn with
sn < b.
We have said that lim sup and lim inf determine the size of the box close to
which sn will stay eventually (sn may venture above lim sup or below lim inf but
by smaller and smaller amounts). So if the box is of size zero, then sn eventually
stays close to some particular number, so it stands to reason that that number is
sn ’s limit.
Proposition. If lim sup sn = lim inf sn = s then sn converges to s.
Proof. For a given ε > 0 we need s − ε < sn < s + ε for all n > N . This is
basically saying that s + ε is an upper bound for {sn |n > N } and s − ε is a lower
bound. Those are true precisely when vN ≤ s + ε and uN ≥ s − ε. But both vN
and uN converge to s, so that holds for large N .
The formal proof is written in reverse.
For any ε there exists N1 such that vN1 < s + ε and exists N2 such that
uN2 > s − ε. Then for N > max{N1 , N2 } we get s + ε > vN ≥ sn ≥ uN > s − ε,
so that |s − sn | < ε, as wanted.
There is also an alternative proof.
Proof. We have un ≤ sn+1 ≤ vn , and as lim vn = lim sn = s by the squeeze
theorem sn+1 converges to s as well, but then so does sn .
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Note that lim sup sn = lim inf sn = ∞ means uN diverge to +∞, so for any
M there is N with uN > M and hence sn ≥ uN < M . and sn diverges to
+∞. Similar result holds for lim sup sn = lim inf sn = −∞. So the Proposition
extends to those cases.
Next time we will show the converse and use these “limit detection” properties of lim inf and lim sup to get an equivalent definition of convergent sequence
without explicitely mentioning the limit!
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