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Would you use Trichotomy?

  1. Nov 3, 2008 #1
    Let [tex] X [/tex] be a metric space, let [tex] a \in X [/tex] be a limit point of [tex] X [/tex], and let [tex] f: X \to \mathbb{R} [/tex] be a function. Assume that the limit of [tex] f [/tex] exists at [tex] a [/tex]. Fix [tex] t \in \mathbb{R} [/tex]. Suppose there exists [tex] r > 0 [/tex] such that [tex] f(x) \geq t [/tex] for every [tex] x \in B_{r}(a) \backslash \{a \} [/tex]; then [tex] \lim_{x \to a} f(x) \geq t [/tex].

    How would you prove this? Would you use Trichotomy?
  2. jcsd
  3. Nov 3, 2008 #2


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    Re: Continuity

    Find a sequence xk such that xk converges to a. Then what can you say about f(xk) for each k?
  4. Nov 3, 2008 #3
    Re: Continuity

    you will get a contradiction pretty easy if lim f(x)<t.
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