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Let u ˆ be a cluster point of { u n }.
which means that cannot be a cluster point of.
Let be a cluster point of and let be the subsequence converging to.
Let û be a cluster point of the sequence ({u_{n}}).
Let be a cluster point of ; we begin to prove that.
Let (x^) be a cluster point of ({x^{k}}) and a subsequence ({ x^{k_{j}}}) converge to (x^).
Similar(47)
(ii), then is a cluster point of.
., or.. Then is a cluster point of.
By Lemma 1.1, is a cluster point of.
It is obvious that a cluster point of { f x n } is a cluster point of { y n }.
This contradicts the assumption that (tilde{u}) is a cluster point of ({u^{k}}).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com