# Fundamental Mathematics Sequences

FUNDAMENTAL MATHEMATICS II

Question 1. Say that a sequence (an) is a Cauchy sequence (named after

the French mathematician Cauchy) if it has the following property:

For every > 0 there is a number M (depending on ) such that

|an - |am < for all n, m >= M.

(1) * Show that the sequences ( 1/n) and (n + 1/2n) and Cauchy sequences, but the sequence (n) is not.

(2) * Write down the definition of what it means for a sequence (an) to be convergent. Show that any convergent sequence is a Cauchy sequence.

(3) Show that every Cauchy sequence in R converges to a limit in R.

(4) * Give an example to show that there can be a Cauchy sequence

(an) of rational numbers that does not converge in Q.

Question 2. In this module we shall build upon the second definition of a

function being continuous that we had in Chapter 4, Definition

1.6. We recall that Definition here.

Definition. Say that the function f : R --> R is continuous at the point

a 2 R if for all > 0 there is a number > 0 with the property

|f(x) - f(a)| < e for all x with |x- a| < s

(1) * Use this Definition to show that the function

(a) f : R --> R given by f(x) = 2x is continuous at the point a 2 R

(b) g : R --> R given by g(x) = x2 is continuous at the point a 2 R

(c) : R --> R given by

s(x) =( 1 if x = 0

( 0 if x 6= 0

is not continuous at the point 0 e R.

(2) Prove that a function f : R ! R is continuous at a point a 2 R in

the sense of the definition above, if and only if it is continuous at a

in the sense that limx-->a f(x) = f(a).

#### Solution Summary

Fundamental mathematics sequences are examined in the solution. It shows that every Cauchy sequence in R converges to a limit in R.