Question

2.20 Let ƒ be a real-valued function defined for every x in the interval 0 ? x ? 1. Suppose there is a positive number M having the following property: for every choice of a finite number of points x1, x2,...,x in the interval 0 ? x ? 1, the sum $|f(x_1) + ... + f(x_n)| ? M$. Let S be the set of those x in 0 ? x ? 1 for which f(x) ? 0. Prove that S is countable. 2.21 Find the fallacy in the following "proof" that the set of all intervals of positive length is countable. Let {x1, x2,...} denote the countable set of rational numbers and let I be any interval of positive length. Then I contains infinitely many rational points x_n, but among these there will be one with smallest index n. Define a function F by means of the equation F(I) = n, if x_n is the rational number with smallest index in the interval I. This function establishes a one-to-one correspondence between the set of all intervals and a subset of the positive integers. Hence the set of all intervals is countable.

          2.20 Let ƒ be a real-valued function defined for every x in the interval 0 ? x ? 1.
Suppose there is a positive number M having the following property: for every choice of
a finite number of points x1, x2,...,x in the interval 0 ? x ? 1, the sum
$|f(x_1) + ... + f(x_n)| ? M$.
Let S be the set of those x in 0 ? x ? 1 for which f(x) ? 0. Prove that S is countable.
2.21 Find the fallacy in the following "proof" that the set of all intervals of positive
length is countable.
Let {x1, x2,...} denote the countable set of rational numbers and let I be any
interval of positive length. Then I contains infinitely many rational points x_n, but among
these there will be one with smallest index n. Define a function F by means of the equation
F(I) = n, if x_n is the rational number with smallest index in the interval I. This function
establishes a one-to-one correspondence between the set of all intervals and a subset of the
positive integers. Hence the set of all intervals is countable.

2.20 Let ƒ be a real-valued function defined for every x in the interval 0 ? x ? 1.
Suppose there is a positive number M having the following property: for every choice of
a finite number of points x1, x2,...,x in the interval 0 ? x ? 1, the sum
|f(x1) + ... + f(xn)| ? M.
Let S be the set of those x in 0 ? x ? 1 for which f(x) ? 0. Prove that S is countable.
2.21 Find the fallacy in the following "proof" that the set of all intervals of positive
length is countable.
Let x1, x2,... denote the countable set of rational numbers and let I be any
interval of positive length. Then I contains infinitely many rational points xn, but among
these there will be one with smallest index n. Define a function F by means of the equation
F(I) = n, if xn is the rational number with smallest index in the interval I. This function
establishes a one-to-one correspondence between the set of all intervals and a subset of the
positive integers. Hence the set of all intervals is countable.

Added by Christina F.

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