A Proof of the Squeeze Theorem for Integrals Using Cauchy Sequences

A Proof of the Squeeze Theorem for Integrals Using Cauchy

Sequences

Spiros Konstantogiannis [email protected]

Abstract. We use the sequential criterion for Riemann integrability to give a proof of the squeeze theorem for integrals using Cauchy sequences. As an auxiliary lemma, we prove a criterion for a sequence to be a Cauchy sequence.

Keywords: Riemann integral; sequential criterion; Cauchy sequence; squeeze theorem.

1 Introduction

Sequential criterion for Riemann integrability

A function

f

: ,

[ ]

a b

®

¡

is Riemann integrable on

[ ]

a b

,

if and only if for every sequence

( )

P

&

_n

of tagged partitions of

[ ]

a b

,

which is such that

lim

_n

0

n®¥

P

=

&

_{, the sequence}

(

(

_;

)

)

n

S f P

&

of

Riemann sums is a Cauchy sequence, and then

lim

(

;

)

b n n a

S f P

f

®¥

=

ò

&

_. Proof

The proof is given in [1].

Auxiliary lemma 1 (a consequence of the definition of limit)

If the sequences

( )

x

_n and

( )

y

_n both converge and

lim

_n

lim

_n

n®¥

x

<

n®¥

y

+

a

, for some

a

Î

¡

, then

for every

e

>

0

there exists

N

Î

¥

such that

x

_n

<

y

_n

+ +

a

e

for all

n N

>

.

Proof

Since both sequences

( )

x

_n and

( )

y

_n converge, then, by linearity, the sequence

(

x

_n

-

y

_n

-

a

)

also converges and

lim

(

_{n n}

)

lim

_n

lim

_n

n®¥

x

-

y

-

a

=

n®¥

x

-

n®¥

y

-

a

(1).

Further, by assumption,

lim

_n

lim

_n

n®¥

x

<

n®¥

y

+

a

, thus n

lim

®¥

x

n

-

lim

n®¥

y

n

- <

a

0

, and by means of (1),

(

)

lim

_{n n}

0

n®¥

x

-

y

-

a

<

(2).

Next, setting

lim

(

_{n n}

)

Besides, since

lim

(

_{n n}

)

n®¥

x

-

y

-

a

=

l

, then, by definition, for every

e

>

0

there exists

N

Î

¥

such

that

x

_n

-

y

_n

- - <

a l

e

for all

n N

>

, from which it follows that

x

_n

-

y

_n

- - <

a l

e

, and then

n n

x

-

y

- < +

a l

e

(4) for all

n N

>

.

Further, adding

e

to both sides of (3) yields

l

+ <

e e

and combining with (4) then yields

n n

x

-

y

- <

a

e

, and thus

x

_n

<

y

_n

+ +

a

e

for all

n N

>

.

We have thus shown that for every

e

>

0

there exists

N

Î

¥

such that

x

_n

<

y

_n

+ +

a

e

for all

n N

>

, which is what we wanted to prove.

Auxiliary lemma 2 (a criterion for a sequence to be Cauchy)

A sequence

( )

x

_n is a Cauchy sequence if and only if for every

e

>

0

there exists

N

Î

¥

and

l

_e

Î

¡

(that depends on

e

), such that

x

_n

- <

l

_e

e

for all

n N

>

.

Proof

Note that, since

l

_e depends on

e

, then

l

_e is not a fixed number, and thus it cannot be derived that

l

_e is the limit of

( )

x

_n .

(i) Let

( )

x

_n be a Cauchy sequence.

Then, by definition, for every

e

>

0

there exists

N

Î

¥

such that

x

_m

-

x

_n

<

e

for all

m n N

,

>

.

The number

N

generally depends on

e

.

Then, choosing

m N

=

+ >

1

N

, we have that

x

_N+1

-

x

_n

<

e

, and since

(

)

1 1 1

N n N n n N

x

₊

-

x

= -

x

₊

-

x

=

x

-

x

₊ , the last inequality reads

x

_n

-

x

_N+1

<

e

for all

n N

>

.

Further, since

N

depends on

e

, then

N

+

1

also depends on

e

, and thus

x

_N+1 depends on

e

too.

Then setting

x

_N+1

º

l

_e, we have that for every

e

>

0

there exists

N

Î

¥

such that

x

_n

- <

l

_e

e

for all

n N

>

.

(ii) We assume that for every

e

>

0

there exists

N

Î

¥

and

l

_e

Î

¡

such that

x

_n

- <

l

_e

e

(1)

for all

n N

>

.

We will show that the sequence

( )

x

_n is a Cauchy sequence.

Further, by the triangle inequality,

x

_m

- + -

l

_e

l

_e

x

_n

³

(

x

_m

-

l

_e

) (

+

l

_e

-

x

_n

)

=

x

_m

-

x

_n , thus

m n m n

x

-

x

£

x

- + -

l

_e

l

_e

x

, and combining with (4) yields

x

m

-

x

n

<

2

e

.

The last inequality holds for every

e

>

0

, thus it also holds for every

e

¢ =

2

e

>

0

, because as

e

takes any positive value,

2

e

also takes any positive value.

We have thus shown that for every

e

¢ >

0

there exists

N

Î

¥

such that

x

_m

-

x

_n

<

e

¢

for all

,

m n N

>

, and then, by definition,

( )

x

_n is a Cauchy sequence.

Finally, combining (i) and (ii) completes the proof.

2 Squeeze theorem

A function

f

: ,

[ ]

a b

®

¡

is Riemann integrable on

[ ]

a b

,

if and only if for every

e

>

0

there exist

functions

g h

_e

,

_e

: ,

[ ]

a b

®

¡

such that

g x

_e

( )

£

f x

( )

£

h x

_e

( )

for all

x

Î

[ ]

a b

,

, and such that

,

g h

_{e e} are both Riemann integrable on

[ ]

a b

,

and

(

)

b a

h

_e

-

g

_e

<

e

ò

.

Proof

(i) Let

f

be Riemann integrable on

[ ]

a b

,

.

For every

e

>

0

, we set

g

_e

º

f

and

h

_e

º

f

, and then we have

g x

_e

( )

=

f x

( )

=

h x

_e

( )

, thus

also

g x

_e

( )

£

f x

( )

£

h x

_e

( )

, for all

x

Î

[ ]

a b

,

, and since, by assumption,

f

is Riemann

integrable on

[ ]

a b

,

, then

g h

_e

,

_e are both Riemann integrable on

[ ]

a b

,

, and also

(

)

0 0

b b

a a

h

_e

-

g

_e

=

=

ò

ò

, because

S

( )

0;

P

&

=

0

for any tagged partition

P

&

of

[ ]

a b

,

, thus taking any

sequence

( )

P

&

_n of tagged partitions of

[ ]

a b

,

which is such that

lim

_n

0

n®¥

P

=

&

_{, we have that}

( )

0;

_n

0

S

P

&

=

, and then

lim

( )

0;

_n

0

n®¥

S

P

=

&

_{, and thus, by the sequential criterion for Riemann}

integrability,

0 0

b a

=

ò

. Since

(

)

0

b a

h

_e

-

g

_e

=

ò

, then

(

)

b a

h

_e

-

g

_e

<

e

ò

for every

e

>

0

.

We have thus shown that for every

e

>

0

there exist functions

g h

_e

,

_e

: ,

[ ]

a b

®

¡

such that

( )

( )

( )

g x

_e

£

f x

£

h x

_e for all

x

Î

[ ]

a b

,

, and such that

g h

_e

,

_e are both Riemann integrable on

[ ]

a b

,

and

(

)

b a

h

_e

-

g

_e

<

e

ò

.

(ii) We assume that for every

e

>

0

there exist functions

g h

_e

,

_e

: ,

[ ]

a b

®

¡

such that

( )

( )

( )

g x

_e

£

f x

£

h x

_e for all

x

Î

[ ]

a b

,

, and such that

g h

_e

,

_e are both Riemann integrable on

[ ]

a b

,

and

(

)

b a

h

_e

-

g

_e

<

e

ò

.

Next, we consider any sequence

( )

P

&

_n of tagged partitions of

[ ]

a b

,

which is such that

lim

_n

0

n®¥

P

=

&

_.

Then, since

g h

_e

,

_e are integrable on

[ ]

a b

,

, the sequential criterion for Riemann integrability implies

that

lim

(

;

)

b n n a

g

_e

S g P

_e ®¥

=

ò

&

_{(1) and} b

_lim

(

_;

)

n n a

h

_e

S h P

_e ®¥

=

ò

&

_(2).

Since both integrals

b a

g

_e

ò

and b a

h

_e

ò

exist, then, by linearity,

(

)

b b b

a a a

h

_e

-

g

_e

=

h

_e

-

g

_e

ò

ò

ò

, and by (1)

and (2),

(

)

lim

(

;

)

lim

(

;

)

b n n n n a

h

_e

g

_e

S h P

_e

S g P

_e ®¥ ®¥

-

=

-ò

&

&

_{, and then the relation} b

₍

₎

a

h

_e

-

g

_e

<

e

ò

reads

(

)

(

)

lim

;

_n

lim

;

_n n®¥

S h P

e

-

n®¥

S g P

e

<

e

&

&

_{, thus}

_lim

(

_;

)

_lim

(

_;

)

n n

n®¥

S h P

e

<

n®¥

S g P

e

+

e

&

&

_.

Then, by the auxiliary lemma 1, there exists

N

Î

¥

such that

S h P

(

_e

;

&

_n

) (

<

S g P

_e

;

&

_n

)

+ +

e e

, thus

(

;

_n

) (

;

_n

)

2

S h P

_e

&

<

S g P

_e

&

+

e

(3) for all

n N

>

.

Besides, since, by assumption,

g x

_e

( )

£

f x

( )

£

h x

_e

( )

for all

x

Î

[ ]

a b

,

, then for every tagged

partition

P

&

of

[ ]

a b

,

, it holds that

S g P

(

_e

;

&

) ( ) (

£

S f P

;

&

£

S h P

_e

;

&

)

.

Let us prove it.

If

{ }

0 m k k

x

₌ and

{ }

1 m k k

t

₌ are, respectively, the partition points and tags of

P

&

, then, by definition,

(

;

)

( )(

1

)

m k k k

S g P

_e

&

=

å

g t

_e

x

-

x

_- ,

( )

;

( )(

₁

)

m k k k

S f P

&

=

å

f t

x

-

x

_- ,

Further, since

g x

_e

( )

£

f x

( )

£

h x

_e

( )

for all

x

Î

[ ]

a b

,

and since

t

_k

Î

[

x

_k-1

,

x

_k

] [ ]

Í

a b

,

for

1, 2,...,

k

=

m

, then

g t

_e

( )

_k

£

f t

( )

_k

£

h t

_e

( )

_k , and multiplying all sides by

x

_k

-

x

_k-1

>

0

(

x

_k

>

x

_k-1

so that the interval

[

x

k-₁

,

x

k

]

is well defined) yields

( )(

k k k 1

)

( )(

k k k 1

)

( )(

k k k 1

)

g t

_e

x

-

x

_-

£

f t

x

-

x

_-

£

h t

_e

x

-

x

_- , for

k

=

1, 2,...,

m

, and adding the

m

inequalities then yields

( )(

₁

)

( )(

₁

)

( )(

₁

)

1 1 1 m m m k k k k k k k k k k k k

g t

_e

x

x

_-

f t

x

x

_-

h t

_e

x

x

-= = =

-

£

-

£

-å

å

å

, thus

(

;

) ( ) (

;

;

)

S g P

_e

&

£

S f P

&

£

S h P

_e

&

, which is what we wanted to prove.

Applying the previous relation, we have

S g P

(

_e

;

&

_n

) (

£

S f P

;

&

_n

) (

£

S h P

_e

;

&

_n

)

for all

n

Î

¥

, thus also

for all

n N

>

, and combining with (3) then yields

S g P

(

_e

;

&

_n

) (

£

S f P

;

&

_n

) (

<

S g P

_e

;

&

_n

)

+

2

e

(4) for all

n N

>

.

We note that we cannot take the limit

e

®

0

, because as

N

depends on

e

, then

N

may tend to

infinity as

e

tends to 0, and then the relation (4) does not actually hold.

To proceed, we will use that the sequence

(

S g P

(

_e

;

&

_n

)

)

converges to

b a

g

_e

ò

(see (1)).

Then, by definition, for every

e

¢ >

0

, thus also for

e

¢ = >

e

0

, there exists

N

₁

Î

¥

such that

(

;

)

b n

a

S g P

_e

&

-

ò

g

_e

<

e

(5) for all

n N

>

₁.

Next, we set

N

₂

º

max

{

N N

,

₁

}

.

Since

N N

,

₁

Î

¥

, then

max

{

N N

,

₁

}

Î

¥

, and thus

N

₂

Î

¥

.

Further, since

max

{

N N

,

₁

}

³

N N

,

₁, then

N

₂

³

N N

,

₁, and thus if

n N

>

₂, then

n N N

>

,

₁, and

thus both (4) and (5) hold for all

n N

>

₂.

By (5), we have

(

;

)

b n a

S g P

_e

g

_e

e

e

- <

&

-

_ò

<

, and adding

b a

g

_e

ò

to all sides yields

(

;

)

b b n a a

g

_e

- <

e

S g P

_e

<

g

_e

+

e

ò

&

ò

_{, thus} b

(

_;

)

n a

g

_e

- <

e

S g P

_e

ò

&

_{(6) and}

(

_;

)

b n a

S g P

_e

&

<

_ò

g

_e

+

e

_(7).

Next, adding

2

e

to both sides of (7) yields

(

;

)

2

3

b n

a

Further, since

e

>

0

, then

- > -

e

3

e

, and adding

b a

g

_e

ò

to both sides yields

3

b b a a

g

_e

- >

e

g

_e

-

e

ò

ò

, i.e.

3

b b a a

g

_e

-

e

<

g

_e

-

e

ò

ò

(9).

Next, combining (6) and (9) yields

3

(

;

)

b b n a a

g

_e

-

e

<

g

_e

- <

e

S g P

_e

ò

ò

&

_{, thus} b

₃

(

_;

)

n a

g

_e

-

e

<

S g P

_e

ò

&

_,

and combining with (4) then yields

3

(

;

) (

;

) (

;

)

2

b

n n n

a

g

_e

-

e

<

S g P

_e

£

S f P

<

S g P

_e

+

e

ò

&

&

&

_{, thus}

(

) (

)

3

;

;

2

b n n a

g

_e

-

e

<

S f P

<

S g P

_e

+

e

ò

&

&

_{, and combining with (8) yields}

(

)

3

;

3

b b n a a

g

_e

-

e

<

S f P

<

g

_e

+

e

ò

&

ò

_{, and adding} b

a

g

_e

-

ò

to all sides then yields

(

)

3

;

3

b n a

S f P

g

_e

e

e

- <

&

-

_ò

<

, thus

(

;

)

3

b n a

S f P

&

-

ò

g

_e

<

e

(10).

The inequality (10) holds for every

e

>

0

and for every

n N

>

₂.

Since the inequality (10) holds for every

e

>

0

, then it also holds for every

e

¢¢ =

3

e

>

0

, because as

e

takes any positive value,

3

e

also takes any positive value.

Also, since

e e

=

¢¢

3

, then the function

g

_e that, by assumption, depends on

e

, depends on

e

¢¢

.

We have thus shown that for every

e

¢¢ >

0

there exists

N

₂

Î

¥

such that

(

;

)

b n

a

S f P

&

-

ò

g

_e¢¢

<

e

¢¢

for all

n N

>

₂, where the integral

b a

g

_e¢¢

ò

is a real number that, generally, depends on

e

¢¢

(because

the function

g

_e¢¢ depends on

e

¢¢

).

Then, by the auxiliary lemma 2, the sequence

(

S f P

(

;

&

_n

)

)

is a Cauchy sequence.

We have thus shown that for any sequence

( )

P

&

_n of tagged partition of

[ ]

a b

,

which is such that

lim

_n

0

n®¥

P

=

&

_{, the sequence}

(

(

_;

)

)

n

S f P

&

is a Cauchy sequence.

3 References

1. Spiros Konstantogiannis, A Sequential Criterion for Riemann Integrability Resulting from Riemann’s Definition of the Integral. academia.edu, 2020, available at

https://www.academia.edu/43876143/A_Sequential_Criterion_for_Riemann_Integrability_Resulting_from_Riema nns_Definition_of_the_Integral

and

https://q4quantum.wordpress.com/2020/08/17/a-sequential-criterion-for-riemann-integrability-resulting-from-riemanns-definition-of-the-integral/

2. Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis. John Wiley & Sons, Inc., Fourth Edition, 2011.