Distinction between "if any" and "if every". Today in my math class I presented a counter example to the theorem:
"if any infinite sequence in X has an adherent point in X, then X is compact."
Let $X=(-1,2)$. Choose $\{X_{n}\}= \frac{1}{n} = \{1,\frac{1}{2}, \frac{1}{3},\cdots \}$.
Then $0$ is an adherent point of $\{ X_{n} \}$ in $X$.
But X is not compact. 
My professor told me that "if any" is synonymous with "if every" in this instance so my counter example doesn't work but I can't see how that is the case. Can anyone give me some insight into how these statements are equivalent? 
Sorry about any poor formatting I am on mobile.

If you are in doubt, sometimes, putting parentheses around logical statements can help make their meaning clearer. In this instance:
if (any infinite sequence in $X$ has an adherent point in $X$) then ($X$ is compact)
What the first bracketed statement essentially means is "choose any infinite sequence in $X$ and it will have an adherent point in $X$". Now, at this point, it should be quite clear that replacing 'any' with 'every' would give you an equivalent statement.
