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.

1. Identify whether the slope is positive, negative, zero or undefined. Please show work or explain answer so that I can understand it
Please see attachment for graph for question 1.
a) Positive
b) Negative
c) Zero
d) Undefined
2. Find the x- and y-intercepts for the equation. Then graph the equation. Please show work.

(a) Prove this operation:
Let {xn} and {yn} be convergent sequences.
The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn
What I attempted was this:
Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y
Let epsilon>0 be given. Find M1 s.t. for

A. what is d, the difference between any 2 terms?
answer:
show work in this space.
b. using the formuls for the nth term of a arithmetic sequence, what is 101st term?
answer:
show work in this space.
c. using the formula for the sum of an arithmetic series, what is the sum of the first 20 terms?
answer:
show work in

Suppose the sequences {a_n}_n and {b_n}_n are both bounded above.
a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n}
b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n)
(all limits are n--> infinity)

1. The cost, c, in dollars of a car rental is 10 + , where m is the number of miles driven. Graph the equation and use the graph to estimate the cost of car rental if the number of miles driven is 34.
A) About 24 dollars
B) About 15 dollars
C) About 36.5 dollars
D) About 19 dollars
2. Graph the two lines x + 2y