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).