Stupid questions of basic analysis

[jessicaw]

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

 

[wayneckm]

I think the argument is, first of all, I assume \{t_{n}\} take values in \mathbb{R}, then, due to the existence of limit, \inf is indeed \min and so it should be > - \infty. Somehow I think it is also an if-and-only-if statement.

Wayne

 

[Office_Shredder]

The existence of the limit does not imply infimum is minimum.

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

 

[wayneckm]

Argh, you are right.

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