Does a statement of the form "for all $X>0$ there exists $x > X$ satisfying some condition" evaluate to "the condition must be true for all $x>0$"?

Question

Question is essentially as the title states. I was inspired by the form which the negation of the Cauchy Criterion takes.

If I have a statement saying "$P$ is only true if for all $X > 0$, there exists an $x > X$ for which $Q$ is true," does this logically evaluate to "$P$ is only true if $Q$ is true for all $x > 0$"?

The original may take the form of $$P\iff \forall X>0\;\exists x\geq X :Q$$

In theory, one may sketch a proof proceeding by induction, where one might say that since for all $X$ there must be an $x > X$ making $Q$ true, $Q$ must be true for $X = 0.5$ and $x = 1$, and $X = 1$ and $x = 2$, etc. Thus $Q$ must be true for all $x$.

Is this logically permissible?

Answer 1

Here is a counterexample. Say $P(x)$ is true only if $x$ is an even integer. ($P$ and $Q$ could even both be “$x$ is even.” ) Then it is true that, for all $X$, there is $x>X$ such that $P(x)$, but it is not true that, for all $X$, $P(X)$.