1.

Given a sequence {x_n}, n = 1,2,... such that

	( xn-xn-2 ) = 0,

prove that

( xn-xn-1)/n = 0.

2.

A closed subset S of		lies in a < x < b.	Show that its projection
on the y-axis is closed.

3.

A game of solitaire is played as follows. After each play, according to the outcome, the player receives either a or b points (a and b are positive integers with a greater than b), and his score accumulates from play to play. It has been noticed that there are thirty-five nonattainable scores and that one of these is 58. Find a and b.

4.

Consider an infinite series whose n-th term is

(1/n),

with the

signs being determined according to a pattern that repeats periodically in blocks of eight. (There are 2⁸ possible patterns of which two examples are:

The first example would generate the series
1 + (1/2) - (1/3) - (1/4) - (1/5) - (1/6) + (1/7) + (1/8) + (1/9) + (1/10) - (1/11) - (1/12) - ... .)

(a)

Show that a sufficient condition for the series to be conditionally convergent is that there be four "plus" signs and four "minus" signs in the block of eight.

(b)

Is this sufficient condition also necessary? (Here "convergent" means "convergent to a finite limit".)

5.

How many zeros does the function f(x) = 2^x - 1 - x² have on the real line? (By a "zero" of a function f, we mean a value x₀ in the domain of f (here the set of all real numbers) such that f(x₀) = 0.)

6.

Show that the integral equation

has at most one solution for f continuous on 0 < = x < = 1, 0 < = y < = 1.