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