I have a question about this logic statement, in $\mathbb{R}$

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> If $x$ is such that $x \ge y+ \varepsilon$ for every $\varepsilon >0$, then $x \ge y$
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That is correct, right? But it calls my attention the "greater than **or** equal". Is this a correct statement?

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> If $x$ is such that $x = y+ \varepsilon$ for every $\varepsilon >0$, then $x = y$
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I think that would be analogous to saying

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> If $x$ is such that $x = \varepsilon$ for every $\varepsilon >0$, then $x = 0$
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but that one last statement looks wrong. Thanks!