show that an infinite sequence converges if and only if it has exactly one accumulation point (limit point)?

Question

please show me how it would be done in an exam. thanks alot

Answer 1

Well, I think the problem is that the theorem is wrong. In fact, neither implication holds. The sequence 1, 1, 1, 1... given by x_n = 1 converges to 1, but ((0, 2)\{1})∩{x_n: n∈ℕ} is empty, so 1 is not an accumulation point, and it is obvious that neither is any other number. Conversely, the sequence x_n = {1/n if n is even, n if n is odd} does not converge, but nonetheless has exactly one accumulation point in ℝ, namely 0.



Edit: No, 1, 1, 1, 1... has infinitely many terms, all of which happen to be the same. There is no law stating that all the terms of a sequence must be distinct, or even that infinitely many of them must be distinct. The specification infinite sequence is to distinguish it from something like (1, 2, 2, 4), which has only four terms.

Answer 2

Let sequence be denoted x_n and limit point, L.



definition of convergence,



abs {x_n - L} < e,



for an arbitrary e (small), and for all n>= N



So we need to show



1) if the sequence converges it has exactly one limit point, and conversely



2) if the sequence has exactly one limit point, it converges.



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1) Suppose the sequence has 2 limits, say L and M



abs {x_n - L} < e

abs {x_n - M} < e



adding: abs {2x_n - (L+M)} < 2e



abs {x_n - (L+M)/2} < e



(L+M) / 2 = L, L+M = 2L, M=L



so only one limit point by contradiction.



2) If it has one limit point, L, it satisfies the equation



abs {x_n - L} < e



and converges by definition

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I think its right, the 1st step you might wanna check

Answer 3

it truly is feasible that our Universe is "embedded" in yet yet another area, and that the extensive Bang would be localized in that area. if so, that area isn't area of our Universe and is unreachable, a minimum of by way of travelling in time or area indoors the stunning-regular way. There would nicely join beginning up place in that hypothetical embedding area. yet from indoors the Universe, there is now no longer any midsection, and no midsection of advance. one way of thinking approximately it particularly is the two-dimensional case of the outdoors of a balloon. if the accomplished Universe have been the outdoors of a balloon, then by way of actuality the balloon inflates, each and every and each little component on the outdoors strikes distant from each and every and each little component else, yet there is now no longer any midsection indoors the Universe itself---there is in ordinary words a center indoors the "embedding area" of three dimensions the region the balloon is residing. yet once you have been a 2-dimensional guy or woman residing on that balloon floor, the three dimensional embedding area would be unreachable. that's truly helpful to to be waiting to come upon the curvature of your Universe, yet you does now now no longer be waiting to locate any midsection. it truly is feasible that the "embedding area" for the Universe does now no longer exist. if so, the geometry would be a four-dimensional version of the balloon floor (a 4-d hyperballoon), even though the embedding area would have not have been given any actuality. it particularly is the stunning-regular theory interior the returned of the extensive Bang form, and individuals carry with that theory by way of actuality there is now no longer any records for any embedding area. If the outdoors of the balloon is all you already comprehend and would desire to comprehend, then the "area" of the Universe, its area of beginning up place, does now no longer exist. If the embedding area exists and obeys Euclidean geometry, it extremely is going to have a minimum of 6 dimensions with the component to hold a curved 4-d merchandise like our Universe. None of this has to do with the assumption of Early Inflation, which says that the advance of the Universe grew to grow to be into very speedy indoors the beginning up.

Answer 4

what do you know about accumulation points( limit points), definition?