数学专业英语2.7-c翻译(2)

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.

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