Young, cheap tuo cha showdown!      Tea #1    Tea #2    Tea #3
Let's see what happens!  And now....  The Reveal  Xia Guan
Yellow Ribbon   Jia Ji Tuo Cha * Menghai Dayi Pu-erh Tea
Mystery Tea From ebay    And there you have it!   Almost Cauchy
Sequence, unfinished business.  I just remembered a question
from beginning analysis that my professor mentioned in class:
"Find a sequence A that is not Cauchy, yet any consecutive terms
A_n, A_{n+1} 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, A, where \forall \; \epsilon > 0 \quad \exists N \in
\mathbb{N} \;  \ni \; \forall n > N \quad |A_n - A_{n+1}| <
\epsilon and \exists \epsilon > 0 \; \ni  \quad \forall  N \in
\mathbb{N} \quad \exists m,n > N \quad |A_m