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Simpler proof that sequence x_n = n does not converge?

  1. Sep 19, 2009 #1
    Hi, I am trying to prove that
    \{n\}_{n\in \mathbb{N}}
    does not converge (based on definition of convergence).

    I can prove this by contradiction saying assume it converges, fix [itex] \epsilon [/itex] , then [itex] x_n < \epsilon + a [/itex] (for [itex] n \ge N [/itex] where N is fixed) (by fundamental theorem of ineq.) but by Archimedean Principle, I can find a natural number that surpasses this bound, i.e. [itex] \exists m , m x_n > \epsilon + a [/itex] which is an element of the sequence [itex]x_n[/itex] which means for some M, [itex] n \ge M \rightarrow x_n > \epsilon + a [/itex] which is a contradiction.

    However this seems like a long complicated proof for a very simple and obvious fact, I was wondering if there is not some easier, more elegant way to prove this that I am missing?

    Thanks for your help.
  2. jcsd
  3. Sep 19, 2009 #2
    How about saying that a convergent sequence is bounded? It is a simple exercise using the triangle inequality.
  4. Sep 19, 2009 #3


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    When you have limited tools at your disposal, that tends to make proofs long and complicated. (and obvious facts tend to be either very easy to prove or very tricky to prove)

    However, I'm not really convinced that your proof is all that long and complicated. It's only what? Three lines long? And the proof boils down to nothing more than:
    If it converges, it would eventually have to stop growing. However, it keeps growing forever. Therefore, it doesn't converge​
    does it not?

    I assert the proof you gave is a direct translation of the above heuristic argument into a rigorous mathematical argument.

    Actually, if you notice, the heuristic argument I gave is full of little missing details that would be very cumbersome to state in natural language. (e.g. a sequence can grow forever but still converge. So I really mean something about how fast it grows) But in this case, those details are very easy to state in the mathematical language!
  5. Sep 20, 2009 #4
    Thanks for the feedback.

    And I guess it isn't that long, it was just in my head as I was writing it out going through the process I was thinking "I can't believe I have to write all this for something that is so obvious."
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