Theorem. Let $\displaystyle D \subset Y$. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number $\displaystyle \epsilon > 0$ such that if $\displaystyle E \subset D$ with $\displaystyle d(E)<\epsilon$ then $\displaystyle E \subset A_{\alpha}$ for some $\displaystyle A_{\alpha}\in C$.
Note that d(E) is the diameter of E.

This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose $\displaystyle \epsilon$ small enough that E is approximately a point?

Theorem. Let $\displaystyle D \subset Y$. Suppose that D is non-empty and sequentially compact. Let C be an open cover of D. Then there is a real number $\displaystyle \epsilon > 0$ such that if $\displaystyle E \subset D$ with $\displaystyle d(E)<\epsilon$ then $\displaystyle E \subset A_{\alpha}$ for some $\displaystyle A_{\alpha}\in C$. Note that d(E) is the diameter of E.
This seems so intuitively obvious to me... or am I not reading it correctly? Can't I just choose $\displaystyle \epsilon$ small enough that E is approximately a point?
Suppose not. If $\displaystyle n\in\mathbb{Z}^+$ there must be a set $\displaystyle B_n$ such that $\displaystyle 0<d(B_n)<\frac{1}{n}$ but $\displaystyle B_n$ is a subset of no open set in the cover.