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Limit definition and infinitely often

  1. Dec 1, 2012 #1
    Limit definition and "infinitely often"

    If we have a sequence of real numbers [itex]x_{n}[/itex] converging to [itex]x[/itex], that means [itex]\forall \epsilon > 0, \exists N [/itex] such that [itex] |x_n - x| < \epsilon, \forall n \geq N.[/itex]

    So, can we say [itex] P (|x_n - x| < \epsilon \ i.o.) = 1[/itex] because for [itex]n \geq N[/itex], [itex] |x_n - x| < \epsilon [/itex] always holds?
  2. jcsd
  3. Dec 1, 2012 #2

    Stephen Tashi

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    Science Advisor

    Re: Limit definition and "infinitely often"

    Is that notation suppose to denote a probability? It doesn't define a probability until you establish a scenario that specifies at least one random variable and its probability distribution. Are you thinking of "picking an x_i at random"? Or did you mean the [itex] x_i [/itex] to be a sequence of real valued random variables instead of a sequence of real numbers? (If the [itex] x_i [/itex] are random variables, you have to use a different definition of limit than the one you gave.)
  4. Dec 1, 2012 #3
    Re: Limit definition and "infinitely often"

    Yes, the notation [itex] P(\cdot) [/itex] was supposed to denote a probability, and I want the [itex] x_i [/itex] to be a sequence of real numbers, not random variables. But I understand what you're saying. I cannot talk about probabilities without having a random variable.
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