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How to equate logical statements

  1. Apr 9, 2014 #1
    1. The problem statement, all variables and given/known data

    We have three statements that describe convergence, but using different variables.
    We are asked to show that all statements are equivalent.

    2. Relevant equations

    Show that

    i. [itex]\forall[/itex]ε > 0 ∃N [itex]\forall[/itex]n ≥ N : |a[itex]_{n}[/itex]-a|< ε
    ii. [itex]\forall[/itex]δ > 0 ∃M [itex]\forall[/itex]n ≥ M : |a[itex]_{n}[/itex]-a|≤ δ
    iii. [itex]\forall[/itex]λ > 0 ∃K [itex]\forall[/itex]n ≥ K : |a[itex]_{n}[/itex]-a|≤ 42λ[itex]^{2}[/itex]

    are equivalent statements.

    3. The attempt at a solution

    I know that I have to somehow reformulate epsilon to include/represent delta and lambda, but don't really have an idea how to proceed.
    It is also confusing to me that 42λ[itex]^{2}[/itex] and δ could be equal to the difference of the sequence and limit, whereas ε is only larger than.
    For i to be equivalent to ii and iii, we would need to exclude this case, or include it for epsilon in i, right? (assuming the sequence and limit are 'the same' in each)

    I'm not interested in getting the answer, but rather an approach as this course will likely be proof-heavy and I really want to get a handle on the techniques.
    I just don't know what I'm 'allowed' to do (for example, I can't simply say N = M = K, or can I? And then, does that really show anything?)

    Thanks for reading.
     
  2. jcsd
  3. Apr 9, 2014 #2

    vela

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    One approach would be to show that (i) implies (ii), (ii) implies (iii), and (iii) implies (i). To show (i) implies (ii). You assume (i) is true and use it to prove (ii) is true.

    Writing proofs for "obvious" things can be tricky because it's hard to figure out exactly what you need to show. You need to be pedantic and point out every little step.
     
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