- #1

jessicaw

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Why [itex]\lim t_n >-\infty\Rightarrow inf\{t_n:n\in\mathbb N\} >-\infty[/itex]?

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In summary, the argument is that if a sequence takes values in the real numbers and has a limit, the infimum is also the minimum and therefore greater than negative infinity. However, this is not always the case and the existence of a limit does not necessarily imply that the infimum is the minimum. It is a general fact that a sequence with a limit is bounded, and this can be proven by showing that only finitely many points can be a distance greater than 1 away from the limit.

- #1

jessicaw

- 56

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Why [itex]\lim t_n >-\infty\Rightarrow inf\{t_n:n\in\mathbb N\} >-\infty[/itex]?

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- #2

wayneckm

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Wayne

- #3

Office_Shredder

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It's a general fact that if a sequence of points has a limit, the sequence is bounded. The proof can be sketched as follows: Only finitely many points can be a distance greater than 1 away from the limit (by the definition of a limit). So a lower bound of the set is either one of the values farther away than 1 from the limit, or one less than the limit is a lower bound

- #4

wayneckm

- 68

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I made a mistake in assuming that the bound can be attained within the finite set in the infimum but indeed it is not necessary true. Thanks.

Wayne

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