D. Monotone Sequences

Vocabulary. A sequence q_k with the property q_k ³ q_k+1 for every whole number k, is said to be a decreasing sequence. A sequence q_k with the property q_k £ q_k+1 for every whole number k, is said to be an increasing sequence. A sequence q_k is said to be monotonic if it is an increasing sequence or a decreasing sequence.

Theorem D.2. [On Decreasing Sequences]

If the infinite sequence q₁,q₂,q₃¼ has the property that for each whole numbers k, q_k ³ q_k+1 ³ Q for some real number Q, a lower bound, then the sequence converges to a limit A ³ Q. The limit A is the greatest lower bound of the set of points in the sequence.

Proof:

First, by mathematical induction, n < j implies q_n ³ q_j.

Second, if there exist a whole number K > 0 such that k ³ K implies q_k = q_K then the limit is given by A = q_K and the sequence takes on at most K distinct values q₁,¼,q_K. Otherwise for every K > 0, there is a k > K such that q_k ¹ q_K. In this case, the range of values assumed by the sequence is an infinite set. The case where the range of values is an infinite set is considered next.

An small assertion: observe A > q_m for some m implies that A > q_m ³ q_n for all n ³ m. This in turn implies A is not a limit point. The interval of length q_m-A centered at A contains at most m elements of the sequence. That A > q_m for some m implies that A is not a limit point of the sequence q_j ³ q_j+1.

According to the Weierstrass theorem, the infinite set formed by the sequence q_j has a limit point A. The contrapositive of the above assertion now implies that the limit point A £ q_p for all whole numbers p. But now for each whole number k, the interval A+[1/2]10^-k to A (with A included) contains infinitely many elements of the sequence. Therefore there is some sequence element q_n with the property that A+[1/2]10^-k ³ q_n ³ A. Therefore p ³ n implies A+[1/2]10^-k ³ q_n ³ q_p ³ A since q_p ³ A and q_n ³ q_p whenever p > n. Finally, since k is arbitrary, we conclude include the sequence q_n converges to a limit A. Moreover, we can say there are no points in the sequence q_n to the left of A.

Theorem D.3 [On Increasing Sequences]

If the infinite sequence q₁,q₂,q₃¼ has the property that for each whole numbers k, q_k £ q_k+1 £ Q for some real number Q, an upper bound, then the sequence converges to a limit B £ Q. The limit B is the least upper bound of the set of points in the sequence.

the Real Analysis appendices of

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7

Limits of Monotonic Sequences

the Real Analysis appendices of

Why Slopes and More Math Volume 3

Printed in Canada ISBN 0-9697564-3-7

Limits of Monotonic Sequences

Why Slopes
and
More Math
Volume 3

Printed in Canada
ISBN 0-9697564-3-7