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** 6.12**
**Definition ( approximates .)**
Let

be a positive number, and let

and

be arbitrary numbers.

I will say that

*approximates with an error smaller than*
if
and only if

**Remark**: If approximates with an error smaller than
, then approximates with an error smaller than ,
since
.

** 6.13**
**Definition (Approximation to decimals.)**
Let

, and let

be real numbers. I will say that

*
approximates with decimal accuracy* if and only if

approximates

with an error smaller than

; i.e., if and only
if

** 6.14**
**Notation.**
If I write three dots (

) at the end of a number written in
decimal notation, I assume that all of the digits before the three
dots are correct. Thus since

I have

, and

with 4 decimal accuracy.

** 6.15**
**Example.**
and

Hence

and

Hence

approximates

with an error smaller than

,
and

approximates

with 2 decimal accuracy.

** 6.16**
**Example.**
We see that

and

but there is no two digit decimal that approximates

with 2 decimal
accuracy.

** 6.17**
**Example.**
Since

we see that

approximates

with 4 decimal accuracy, even though
the two numbers have no decimal digits in common.
Since

we see that

does not approximate

with 4 decimal
accuracy, even though the two numbers have four decimal digits in common.

** 6.18**
**Theorem (Strong approximation theorem.)** *Let and be real numbers. Suppose that for every positive number
, approximates with an error smaller than . Then
.
*
Proof: Suppose that approximates with an error smaller than
for every positive number . Then

Hence

i.e.,
. But
, so it follows that
, and consequently ; i.e., .

** Next:** 6.3 Convergence of Sequences
** Up:** 6. Limits of Sequences
** Previous:** 6.1 Absolute Value
** Index**
Ray Mayer
2007-09-07