Fill-in-the-blanks 2. Complete the following proof of the squeeze theorem (sand- wich principle, lemma of two policemen). Claim: If three sequences (n)neN, (bn)neN, and (cn)neN of real num- bers satisfy n bn c, starting from some index, and if lim an= lim cn = R,
then the sequence (bn)neN also converges, and limn b, = Proof. Since the beginning of a sequence affects neither the conver- gence nor the limit of the sequence, we may assume that an b, c, holds for all n N. We will show that limn- b, = . Let > 0. We must show that |bn | < from some index on. Idea: the expression bn - has to be estimated from both direc- tions; one relying on the sequence (an)neN, and the other on the se- quence (cn)neN. (Draw a figure!)
Since limnn = and > 0, there exists an n N such that
for all n n
Since limn cn = and > 0, there exists an n N such that
for all
Now choose
n=
With this choice, for any n > n we have n n. Therefore we get -bn<-an|3=an|< the leftmost inequality holds by virtue of the assumption , b,)
bn-cn-
(the leftmost inequality holds by virtue of the assumption bn cn) The above inequalities imply that |bn-|< for all n n. We have thus proved the claim.
