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Math Help - Cauchy Criterion for Function limit

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    Cauchy Criterion for Function limit

    So my lecturer proved the Cauchy criterion for function limits, however I found his proof to be very long, he did it with Heine (function convergence with sequence) instead I wrote my own version of the proof , however I don't know if it is correct, I would appreciate if someone can verify this proof, thank you!
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    Re: Cauchy Criterion for Function limit

    Quote Originally Posted by davidciprut View Post
    So my lecturer proved the Cauchy criterion for function limits, however I found his proof to be very long, he did it with Heine (function convergence with sequence) instead I wrote my own version of the proof , however I don't know if it is correct, I would appreciate if someone can verify this proof, thank you!
    But you are not done. You only proved it one direction: If l exists then the function has the Cauchy property. That is the easy direction. Now prove that if the function has the Cauchy property then l exists. That requires some equivalence to the Heine-Borel theorem.
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    Re: Cauchy Criterion for Function limit

    You are right, I just noticed that it's ''if and only if'' proof, so . I will prove the other direction too, and put it here, Thanks !
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