数学专业英语2.7-c翻译

翻译

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

hand ,we have

f(n) f(n+1) for all n 1 f(n). if, on the other

We call the sequences decreasing and write f(n). .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

f(n) M 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 f(n)and let L denote the least upper bound of

f(n) L the set of function values.then for all n, and we shall

翻译

prove that the sequence converges to L.

Choose any poitive number .since

be an upper bound for all numbers

L f(N) L0 L f(n) f(n)cannot ,we must have f(N) f(n) for some N.If n N we have

for all n N since f(n).Hence we haveL f(N) L , as illustrate in

0 L f(n) Figure 2-7-1. Form these inequalities we find that

for all n N

And this means that the sequences converges to L,as asserted.

If f(n),the proof is similar, the limit in this case being the greatest lower bound of the set of function values.

翻译

单调的实数序列