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Stupid questions of basic analysis

  1. Oct 13, 2010 #1
    Why [itex]\lim t_n >-\infty\Rightarrow inf\{t_n:n\in\mathbb N\} >-\infty[/itex]?
     
  2. jcsd
  3. Oct 14, 2010 #2
    I think the argument is, first of all, I assume [tex] \{t_{n}\} [/tex] take values in [tex] \mathbb{R} [/tex], then, due to the existence of limit, [tex] \inf [/tex] is indeed [tex] \min [/tex] and so it should be [tex] > - \infty [/tex]. Somehow I think it is also an if-and-only-if statement.

    Wayne
     
  4. Oct 14, 2010 #3

    Office_Shredder

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    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
     
  5. Oct 14, 2010 #4
    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
     
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