SEQUENCES
(ADDITIONAL NOTES TO STEWART §11.1)
The Archimedean Property
For each x > 0 in R there exists an N ∈ N such that N > x.
Hence there also exists for each x > 0 an N ∈ N such that N1 < x.
Theorem A
If the limit of a sequence of real numbers exists, then it is unique.
Proof. Suppose that an → a as n → ∞ and an → b as n → ∞, and also suppose that a 6= b.
Let = 12 |a − b|. Then > 0. Since an → a, there exists an N1 ∈ N such that if n > N1 , then
|an − a| < and since an → b, there exists an N2 ∈ N such that if n > N2 , then |an − b| < . Now
let n > max{N1 , N2 }. Then |an − a| < and |an − b| < . But then it follows that
2 = |a − b| ≤ |a − an | + |an − b| < 2
— contradictory. Hence a = b. Theorem B
A convergent sequence of real numbers is bounded.
Proof. Suppose that an → a as n → ∞. Then (consider = 1) there exists an N ∈ N such
that if n > N , then |an − a| < 1, and hence |an | ≤ |an − a| + |a| < 1 + |a| for all n > N . Let
r = max{|a1 |, |a2 |, . . . , |aN |, 1 + |a|}. Then |an | ≤ r for all n ∈ N, so that {an } is bounded. Theorem C
Suppose that {cn } is a sequence in the domain of a real function f , lim cn = c and f is continuous
n→∞
at c. Then lim f (cn ) = f (c).
n→∞
Prove this.
The Monotone Sequence Theorem
1. Suppose that {an } is an increasing sequence. Then {an } is bounded above if and only if {an }
converges, in which case
lim an = sup{an : n ∈ N}.
n→∞
2. Suppose that {an } is a decreasing sequence. Then {an } is bounded below if and only if {an }
converges, in which case
lim an = inf{an : n ∈ N}.
n→∞
1
Proof.
1. Suppose that {an } is an increasing sequence which is bounded above. Then S = {an : n ∈ N} is
bounded above, so that L = sup S exists, and an ≤ L for all n ∈ N. We show that lim an = L:
n→∞
Let > 0. It follows from Theorem 10 (Improper Integrals: Additional Notes) that there exists
an N ∈ N such that L − < aN . Since {an } is increasing, aN ≤ an for all n ≥ N . Hence it
follows that
L − < aN ≤ an ≤ L < L + ,
so that |L − an | < , for all n ≥ N . Consequently lim an = L.
n→∞
Conversely, if {an } is an increasing sequence which converges, then it follows from Theorem B
above that {an } is bounded, so that {an } is bounded above.
2. similar. 2