Notes on Upper and Lower Limits
Define the extended real line as R=R∪ {−∞} ∪ {∞}.
Theorem 1. Let {an}be a sequence of real numbers, and a∈R. Then the following
statements are equivalent:
(1) ais the infimum of extended real numbers a0which are the limits of subse-
quences of {an};
(2) For every > 0: (a) there is a Nsuch that for all natural n≥N,an> a−,
and (b) for an infinite number of indices n,an< a +.
(3) a= sup
n
inf
k≥n
ak.
If any (and hence, all) of properties (1)–(3) hold, ais called the lower limit, or
limit inferior of a sequence {an}:
a= lim an.
Dually, the following theorem holds:
Theorem 2. Let {an}be a sequence of real numbers, and a∈R. Then the following
statements are equivalent:
(1) ais the supremum of extended real numbers a0which are the limits of sub-
sequences of {an};
(2) For every > 0: (a) there is a Nsuch that for all natural n≥N,an< a+,
and (b) for an infinite number of indices n,an> a −.
(3) a= inf
nsup
k≥n
ak.
If any (and hence, all) of properties (1)–(3) hold, ais called the upper limit, or
limit superior of a sequence {an}:
a= lim an.
Upper and lower limits have the following properties:
(1) lim(−an) = −lim an;
(2) inf an≤lim an≤lim an≤sup an;
(3) lim an= lim anif and only if lim anexists. In this case,
lim an= lim an= lim an.
(4) lim an+ lim bn≤lim(an+bn)≤lim an+ lim bn, provided the right and left
sides are not of the form ∞−∞.
(5) if an≥0 and bn≥0 then
lim anlim bn≤lim(anbn)≤lim anlim bn
provided the product on the right is not of the form 0 · ∞.
Problem 1. Find lim(−1)nn
n+1 and lim(−1)nn
n+1 .
1