3.1.3 Definition

Sequnce X = (xn) on R are said to converge to x ϵ R, or x is said to be the limit of (xn), if every 𝛆> 0 there is a natural number K (𝛆) as for all n ≥ K (𝛆), the limit xn meets | xn – x | between xn and x is less than 𝛆 for all n ≥ K (𝛆) = K.

If the sequence has a limit, then the sequence is convergent, if there is no limit the sequence is divergent.

Note: the notation K (𝛆) is used to emphasize that the choice of K depends on the value 𝛆. However, sometimes it’s easier to write K than K (𝛆). In many cases, the value of “small” actually contains a value of “large” K to guarantee distance | xn – x | between xn and x is less than 𝛆 for all n ≥ K (𝛆) = K.

When a sequence has an x limit, we denote it with

Lim X = x or lim (xn) = x

Sometimes the symbol xn ⟾ x is used, which signifies the intuitive idea that the value of xn approaches the value of x as n ⟾ ∞.

3.1.4 uniqueness of the sequence limit in R has at least one limit

Proof :

Let’s say x ’and x’ ’are both limits of (xn). for every 𝛆> 0 there is a K, i.e. | xn – x ’| <𝛆 / 2 for all n ≥ K ’, and there is K’ ’i.e. | xn – x ” | <𝛆 / 2 for all n ‘K’ ‘. Given K is greater than K ‘and K’ ‘. Then for n ≥ K we use the triangle inequality to be obtained

| x’-x ’’ | = | x ’- xn + xn – x’ ’|

≤ | x ’- xn | + | xn – x’ ’| <𝛆 / 2 + 𝛆 / 2 = 𝛆

Because 𝛆> 0 says arbitrarily positive, summed up x ’- x’ ’= 0

For xϵR and 𝛆> 0, remember that the 𝛆-neighborhood of x is the set

V𝛆 (x) = {u ϵ R: | u – x | <𝛆}

Because V𝛆 (x) is equivalent to | u – x | <𝛆, the definition of convergent sequence can be formulated in terms of neighborhoods.

3.1.5 theorem X = (xn) given is a sequence of real numbers, given x ϵ R. The following statement is equivalent.

1. X converges to x.
2. For every 𝛆> 0, there are natural numbers K such that for all n ≥ K, satisfying xn satisfies | xn – x | <𝛆.
3. For each 𝛆> 0, there is the original say K so for all n ≥ K, satisfy x – 𝛆 <xn <x + 𝛆.
4. For every 𝛆-neighborhood V𝛆 (x) in x, there are natural numbers K as for all n ≥ K, the condition xn has V𝛆 (x).

Proof: the equivalent of a and b is only a definition. Equivalents to b, c, and d follow the implications

| u – x | <𝛆 ⟺ – 𝛆 <u – x <𝛆 ⟺ x – 𝛆 <u <x + 𝛆 ⟺ u ϵ V𝛆 (x).

With neighboring language, one can describe the convergence of the line X = (xn) to the number x by saying: for each 𝛆-neighborhood V𝛆 (x) for x, all but the finite number in the X condition belongs to V𝛆 (x). finite numbers on the condition that there may not be any 𝛆-neighborhoods are the conditions x1, x2, …, xk-1.

3.1.7 example

Sequence (0, 2, 0, 2, …, 0, 2, …) does not converge to 0. If player A confirms 0 is the limit of the sequence, he loses K (𝛆) game when player B gives him a value of 𝛆 < 2 More definitely, player B gives player A the value of t0 = 1. Then it is not important player A chooses the number K, the response will not meet the requirements, for player B will respond by choosing an even number n> K. then the value of xn = 2 that | xn – 0 | = 2> 1 = t0. Thus the number 0 is not the limit of the sequence.