Advanced book on Mathematics Olympiad

110 3 Real Analysis

329.Show that if the series

∑

anconverges, where(an)nis a decreasing sequence, then

limn→∞nan=0.

The following fixed point theorem is a direct application of Cauchy’s criterion for

convergence.

Theorem.LetXbe a closed subset ofRn(or in general of a complete metric space)

andf :X→Xa function with the property that‖f(x)−f(y)‖≤c‖x−y‖for any

x, y∈X, where 0 <c< 1 is a constant. Thenfhas a unique fixed point inX.

Such a function is called contractive. Recall that a set is closed if it contains all its

limit points.

Proof.Letx 0 ∈X. Recursively define the sequencexn=f(xn− 1 ),n≥1. Then

‖xn+ 1 −xn‖≤c‖xn−xn− 1 ‖ ≤ ··· ≤cn‖x 1 −x 0 ‖.

Applying the triangle inequality, we obtain

‖xn+p−xn‖≤‖xn+p−xn+p− 1 ‖+‖xn+p− 1 −xn+p− 2 ‖+···+‖xn+ 1 −xn‖

≤(cn+p−^1 +cn+p−^2 +···+cn)‖x 1 −x 0 ‖

=cn( 1 +c+···+cp−^1 )‖x 1 −x 0 ‖≤

cn

1 −c

‖x 1 −x 0 ‖.

This shows that the sequence(xn)nis Cauchy. Its limitx∗satisfiesf(x∗)=limn→∞f(xn)

=limn→∞xn=x∗; it is a fixed point off. A second fixed pointy∗would give rise to

the contradiction‖x∗−y∗‖=‖f(x∗)−f(y∗)‖≤c‖x∗−y∗‖. Therefore, the fixed

point is unique. 

Use this theorem to solve the next three problems.

330.Two maps of the same region drawn to different scales are superimposed so that the

smaller map lies entirely inside the larger. Prove that there is precisely one point

on the small map that lies directly over a point on the large map that represents the

same place of the region.

331.Lettandbe real numbers with||<1. Prove that the equationx−sinx=t

has a unique real solution.

332.Letcandx 0 be fixed positive numbers. Define the sequence

xn=

1

2

(

xn− 1 +

c

xn− 1

)

, forn≥ 1.

Prove that the sequence converges and that its limit is

√

c.