I just remembered a question from beginning analysis that my professor mentioned in class:
"Find a sequence that is not Cauchy, yet any consecutive terms will become arbitrarily close."
It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.
We seek a sequence, , where and . (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)
The natural log function is the most obvious answer. For large vales of n is small. That is, This is true becuase after taking EXP of both sides we have:
Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let then, if we choose some and let (so, ) then
NOT Cauchy at all!
Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? ? So, I guess this is really asking when while . It is necesairy but not sufficent for and , both will have the x-axis as a horizontal asymptote.
Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.