1. Epsilon-Related Proofs in Real Analysis
Allan Zamierowski
11 June 2013
What are Epsilon-Related Proofs?
In real analysis, many concepts require the idea of observing how a function or a sequence
behaves at certain finite numbers or when it tends toward infinity. Unfortunately, examining
these functions or sequences can be difficult if they behave strangely at certain numbers (in-
finity is even worse because it is a concept rather than a number, so it is difficult to describe
what happens AT infinity), so being able to definitively explain what happens at these spots
by observing what happens when we get very close to them is extremely valuable and is
often used for proofs in real analysis. Epsilon-related proofs essentially allow us to take a
very small number > 0 and use it to get very close to some point of interest to see what
happens at that point. Perhaps a better way to explain is to take some number a ∈ R that
we are interested in and look at the interval (a − , a + ), the epsilon-neighborhood around
a, V (a). Observing what happens in this very small interval allows to tell what happens
at a and how a is related to the real numbers nearby. If it still does not make sense, do
not worry, this concept can be confusing at first, especially without concrete examples. We
will look at definitions of certain concepts that rely on this idea as well as related proofs to
(hopefully) make more sense of things.
Convergence of a Sequence
Definition of a Convergent Sequence
A sequence (an) converges to a real number a if, for every > 0, there exists an N ∈ N such
that whenever n ≥ N, it follows that |an − a| < .
Essentially, if we can take a sequence and get it extremely close to some real number as
the number of terms gets large, we can say that it converges to that real number. When we
pick N large enough, |an − a| will always be smaller than the we are given. To see how
this definition works, we will use it with a concrete example to prove the convergence of a
sequence.
Example
Show
lim
n + 1
3n
=
1
3
.
Following our definition, (an) is the sequence (n + 1)/3n and a = 1/3. Then for any > 0,
we want to show that for N sufficiently large, n ≥ N implies
1

2. n + 1
3n
−
1
3
< .
Notice that if we manipulate this inequality through basic algebra, we have
n >
1
3
.
Then let > 0 be given. Choose N > 1/(3 ). Then for any n ≥ N we have
n ≥ N >
1
3
,
which implies
n + 1
3n
−
1
3
< .
Open and Closed Sets
Definition of an Open Set
A set O ⊆ R is open if for all points a ∈ O there exists an epsilon-neighborhood V (a) ⊆ O.
Definition of a Limit Point
A point x is a limit point of a set A if every epsilon-neighborhood V (x) of x intersects the
set A in some point other than x.
Definition of a Closed Set
A set F ⊆ R is closed if it contains its limit points.
For a set to be open, we need to be able to take any point in the set and create at least
one epsilon-neighborhood around that point that is contained by the set. So for some set
O ⊆ R to be open, if we take any point a ∈ O there must exist a respective > 0 such that
(a − , a + ) ⊆ O
For a set to be closed, every limit point of that set must be an element of the set. A limit
point is any point x that is so close to another point in the set that no matter how small
epsilon is, the set (x − , x + ) always intersects the original set at some point other than
x itself. A set does not necessarily contain its limit points, but if it does it is a closed set.
Also, it is possible for a set to be both open and closed (this is sometimes referred to as the
clopen set). The set R is an example of a set that is open and closed.
2

3. Continuity
Definition of Continuity at a Point
A function f : A → R is continuous at a point c ∈ A if, for all > 0, there exists a δ > 0
such that whenever |x − c| < δ for x ∈ A, it follows that |f(x) − f(c)| < .
This definition basically tell us that no matter how close f(x) gets to f(c), we can always
get x really close to c as long as f is continuous at c. Imagine sketching a function really
close to a certain point c at which the function is continuous from either the positive or
the negative side. No matter how close our function gets to f(c), we will never have to lift
our pencil as we get extremely close to c. It is also important to know that if a function is
continuous at every point on an interval, then it is said to be continuous on that interval.
Example
Show that any linear function of the form ax + b is continuous at any point c ∈ R.
Proof. In order to show that ax + b is continuous at c, we want to show that for all
> 0 there exists a δ > 0 such that |x − c| < δ for x ∈ R implies |(ax + b) − (ac + b)| < .
Notice
|(ax + b) − (ac + b)| = |ax − ac| = |a||x − c|.
Let > 0 be given. Choose δ = /|a|. Then we have
|x − c| <
|a|
=⇒ |a||x − c| < =⇒ |(ax + b) − (ac + b)| < .
3