Econ 204 Section 2
GSI: Hui Zheng
Key Words
Deduction, Contraposition, Contradiction, Induction, Binary Relation, Equivalence Relation,
Equivalence Class, Cardinality, Bijection, Numerically Equivalent
Section 2.1 Methods of Proof
°
Deduction
To prove
A
)
Z
, deduction goes like
A
)
B
) ± ± ± )
Y
)
Z
°
Contraposition
To prove
A
)
Z
, contraposition is to prove
:
Z
) :
A:
It goes like
:
Z
) :
Y
) ± ± ± )
:
B
) :
A
°
Induction
A typical structure of proof is
For
n
= 0
(or other initial value), show that the statement is true. This is the base step.
For
n
=
k;
suppose that the statement is true. This is the inductive hypothesis.
For
n
=
k
+ 1
;
use what we get from the inductive hypothesis to show that the statement
holds for the case of
n
=
k
+ 1
Conclude that the statement is true for all
n
.
°
Contradiction
To prove
A
)
Z
by contradiction, we °rst suppose
Z
is not true. Then we check whether
it leads to results that contradict with
A
or what we get from
A
.
Contraposition can be regarded as a special case of contradiction. Contraposition means
:
Z
)
:
A
so that we get results that contradicts with
A
.
°
De°nition Convergence of a Sequence of Real Numbers in Euclidean Metric Space.
A sequence of real numbers
f
x
n
g
converges to a real number
x
if
8
" >
0
9
N
(
"
)
2
N
, for all
n > N
(
"
)
) j
x
n
²
x
j
< "
. We denote it as
x
n
!
x
or
lim
n
!1
x
n
=
x:
The following two examples will use this de°nition. We are going to learn the general de°n
ition of convergence of sequences in metric spaces in Lecture 3. However, they are pretty
similar.
Example 2.1.1
Prove the following statement by deduction:
f
x
n
g
is a sequence of real numbers. If
lim
n
!1
x
n
=
x >
0
;
then there exists
N
2
N
such
that
n > N
)
x
n
>
0
.
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 Summer '08
 ANDERSON
 Equivalence relation, Binary relation, Transitive relation, contraposition

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