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Necessary And Sufficient Number Metric Space Finite Inequality For Real Numbers Real Number Property Sequence Positive Decimal Expansion Decimal Place Decimal Fraction

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Limit Of Real Number Sequence

An endless real number sequence

$\displaystyle a_1,\,a_2,\,a_3,\,\ldots$

(1)

has the real number

as its limit, if the distance between

and

can be made smaller than an arbitrarily small positive number $\varepsilon$ by chosing the ordinal number

sufficiently great, i.e. greater than a number

(the size of which depends on the value of $\varepsilon$ ); accordingly

$\displaystyle \vert L-a_n\vert < \varepsilon$ when $\displaystyle \quad n > N.$

Then we may denote

$\displaystyle \lim_{n\to\infty}a_n \;=\; L$

(2)

or equivalently

$\displaystyle a_n \to L$ as $\displaystyle \quad n \to \infty.$

(3)

Remark 1. One should not think, that when $n = \infty$ . The symbol “ $\infty$ ” represents no number, one cannot set it for the value of . It's only a question of allowing to exceed any necessary value.

Example 1. Using the notation (2) we can write a result

$\displaystyle \lim_{n\to\infty}\frac{2n}{n\!+\!1} \;=\; 2.$

It's a question of that the real number sequence

$\displaystyle \frac{2}{2},\;\frac{4}{3},\;\frac{6}{4},\;\ldots$

has the limit value 2 (e.g. the nine hundred ninety-ninth member $\frac{1998}{1000} = 1.998$ is already “almost” 2!). For justificating the result, let $\varepsilon$ be an arbitrary positive number, as small as you want. Then

$\displaystyle \left\vert 2-\frac{n}{n\!+\!1}\right\vert = \left\vert\frac{2n\!+... ...ert = \left\vert\frac{2}{n\!+\!1}\right\vert = \frac{2}{n\!+\!1} < \varepsilon,$

(4)

when

is chosen so big that

$\displaystyle n > \frac{2}{\varepsilon}\!-\!1.$

(5)

The condition (5) is obtained from (4) by solving this inequality for

. In this case, we have $N = \frac{2}{\varepsilon}\!-\!1$ .

Example 2. The so-called decimal expansions, i.e. endless decimal numbers, such as

$\displaystyle 3.14159265\ldots \,=\, \pi, \quad 0.636363\ldots, \quad 0.99999\ldots,$

(6)

are, as a matter of fact, limits of certain real number sequences. E.g. the last of these is related to the sequence

$\displaystyle 0.9,\;0.99,\;0.999,\;\ldots$

(7)

which may be also written as

$\displaystyle 1\!-\!\frac{1}{10},\; 1\!-\!\frac{1}{10^2},\; 1\!-\!\frac{1}{10^3},\;\ldots$

The limit of (7) is 1. Actually, if $\varepsilon > 0$ , the distance between 1 and the $n^\mathrm{th}$ member of (7) is

$\displaystyle \left\vert 1-\left(1\!-\!\frac{1}{10^n}\right)\right\vert = \frac{1}{10^n} < \varepsilon,$

when $10^n > \frac{1}{\varepsilon}$ , i.e. when $n > -\log_{10}\varepsilon = N$ .

The endless decimal notations (6) and others are, in fact, limit notations -- no finite amount of decimals in them suffices to give their exact values.

Remark 2. In both of the above examples, no of the sequence members was equal to the limit, but it does not need always to be so; thus for example

$\displaystyle \lim_{n\to\infty}\frac{1\!+\!(-1)^n}{2n} = 0$

and every other member of the sequence in question is 0.

Infinite limits of real number sequences

There are sequences that have no limit at all, for example $1,\,-1,\,1,\,-1,\,1,\,-1,\,\ldots$ . Some real number sequences (1) have the property, that the member may exeed every beforehand given real number if one takes greater than some value (which depends on ):

$\displaystyle a_n > M$ when $\displaystyle \quad n > N.$

Then we write

$\displaystyle \lim_{n\to\infty}a_n \;=\; \infty.$

Similarly, the sequence (1) may be such that for each positive

there is

such that

$\displaystyle a_n < -M$ when $\displaystyle \quad n > N,$

and then we write

$\displaystyle \lim_{n\to\infty}a_n \;=\; -\infty.$

E.g.

$\displaystyle \lim_{n\to\infty}n^2 \;=\; \infty, \quad \lim_{n\to\infty}(1\!-\!n) \;=\; -\infty.$

1	Metric Space
1	Inequality For Real Numbers
1	Real Number
1	Positive
1	Property
1	Sequence
1	Finite
1	Necessary And Sufficient
1	Number
2	Decimal Place
2	Decimal Expansion
6	Decimal Fraction