Monotonic sequences of real numbers
A sequences {f(n)} is said to be increasing if
f(n)f(n+1) for all n 1≤≥
We indicate this briefly by writeing
(n)f . if, on the other
hand ,we have
(n)f(n+1) for all n 1 f ≥≥ We call the sequences decreasing and write (n)f . .A sequences is called monotonic if it is increasing or if it is decreasing.
Monotonic sequences are pleasant to work with because their convergence or divergence is particularly easy to determine. In fact ,we have the following simple criterion.
Note:A sequences {f(n)} is called bounded if there exists a positive number M such that
(n ) M f ≤ for all n. A sequences that is not bounded is called unbounded.
It is clear that an unbounded sequence cannot converge. Therefore, all we need to prove is that a bounded monotonic sequences must converge.
Assume (n)f and let L denote the least upper bound of
the set of function values.then (n) f L ≤ for all n, and we shall
prove that the sequence converges to L.
Choose any poitive number ε.since
0(n)L f ε≤-< cannot be an upper bound for all numbers
f(n),we must have (N)L L f ε-≤ for some N.If n N ≥ we have
(N)f(n)f ≥ since (n)f .Hence we have (N)L L f ε-≤ for all n N ≥, as illustrate in
Figure 2-7-1. Form these inequalities we find that
0(n)L f ε≤-< for all n N ≥
And this means that the sequences converges to L,as asserted.
If (n)f ,the proof is similar, the limit in this case being the greatest lower bound of the set of function values.
单调的实数序列
如果一个序列是增加
≤≥
f(n)f(n+1) for all n1
我们指出这个简要写全方位。

另一方面,如果我们有
f≥≥
(n)f(n+1) for all n 1
我们称之为序列减少和写作。

一个序列称为单调增加或者减少。

单调序列是愉快的工作,因为他们的收敛或发散特别容易确定。

事实上,我们有以下简单的准则。

f≤存在这样一个正注意:一个序列{f(n)}被称为有限如果(n)M
数n。

不是有界被称为无限的序列。

很明显,一个无限序列不可能达成一致。

因此,我们需要证明的是,单调有界序列必须收敛。

假设,(n)
f让L表示的一组函数的最小上界值。

然后对所有≤n,我们将证明序列收敛于L。

f L
(n)
正整体解选择任何号码。

因为不能一个上界为所有数字,我们必须有一些N。

如果我们有。

因此我们有,说明如图2-7-1所示。

我们发现0(n)
L fε
≤-<所有n N
≥
这意味着序列收敛于L,断言。

证明是类似的,如果(n)
f限制在这种情况下的最大下界函数值的设置。