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iogann1982 [59]
3 years ago
13

Show that a sequence {sn} coverages to a limit L if and only if the sequence {sn-L} coverages to zero.

Mathematics
1 answer:
Andreyy893 years ago
3 0

Let {s_n}_{n\in\Bbb N} be a sequence that converges to L. This means for any \varepsilon>0, there is some N such that |s_n-L| for all n>N. From this inequality we see that |(s_n-L)-0|, so it follows that s_n-L\to0.

On the other hand, let {s_n-L} be a sequence that converges to 0. This means |(s_n-L)-0| for all large enough n, and we get the simpler inequality for free, |s_n-L|, so it follows that s_n\to L.

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