1
Nested Quantifiers
1.1
Welcome
Welcome to the seventh module in our course on proofs, “Nested Quantifiers.”
Before going on,
please do the assigned reading.
1.2
Introduction
Almost everyone who takes a calculus course encounters the notion of a limit. When we write
lim
x
→
a
f
(
x
) =
L
we informally mean that we can make the values of
f
(
x
) arbitrarily close to
L
by taking
x
sufficiently
close to, but not equal to
a
. But formally we need to be more explicit about what “arbitrarily” and
“sufficiently” mean. That leads to the infamous

δ
definition of a limit.
The
limit
of
f
(
x
), as
x
approaches
a
, equals
L
means that for every
>
0 there exists a
δ >
0
such that
0
<

x

a

< δ
⇒ 
f
(
x
)

L

<
The arbitrarily close is captured by the choice of epsilon, and the fact that

f
(
x
)

L

must be
less than epsilon. The sufficiently close, is captured by delta, and the fact that

x

a

must be less
than delta, and must also imply that

f
(
x
)

L

is less than epsilon.
Though you may not understand the definition, the important observation for this module is
that the definition contains two quantifiers. We need to learn how to understand statements with
nested quantifiers.
There are really two basic principles to keep in mind.
1. Process quantifiers from left to right.
2. Use existing construction, choose and specialization techniques as you proceed from left to
right.
Moving from left to right is important. The order of quantifiers matters. For example, consider
the following statement about the integers:
∀
x
∃
y, y > x.
Translated into prose, this statement can be read as “Given any integer x, there exists a larger
integer
y
.” This is a true statement. Now let’s make a small modification. We will just change the
order of the quantifiers. Our new statement is
∃
y
∀
x, y > x.
A translation for this statement would be “There exists an integer
y
which is larger than all integers”.
A very different, and false, statement.
1
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1.3
Understanding Nested Quantifiers
Let’s carefully parse our definition of limit. We must proceed from left to right, and in keeping with
our understanding of quantifiers, we must identify the object, the universe of discourse, the property
and the something that happens. The definition is:
For every
>
0 there exists a
δ >
0 such that
0
<

x

a

< δ
⇒ 
f
(
x
)

L

<
The first quantifier encountered is a universal quantifier in the form “for every”. For this quan
tifier we have:
Object:
Universe of discourse:
R
Certain property:
>
0
Something happens:
there exists a
δ >
0 such that 0
<

x

a

< δ
⇒ 
f
(
x
)

L

<
The “something that happens” contains a nested existential quantifier with the following com
ponents:
Object:
δ
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 Spring '08
 ANDREWCHILDS
 Calculus, Continuous function, Philosophy of mathematics, Universal quantification, certain property

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