If we have a real line and $X$ is any subset of it, let $Y$ be a set such that $X\subseteq Y \subseteq \bar{X}$. Prove that $\bar{X}=\bar{Y}$

My Attempt
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We can use definition and show that $\bar{X} $ is in $\bar Y$. For example, let's say we have an element $z$ is $\bar Y$ then for every $\epsilon$ we have to look for an element $x$ in $X$ such that $$|x-z|\leq \epsilon$$ $z$ is in $\bar Y$ so we can have a $y$ in $Y$ such that $$|x-y|\leq \epsilon/2$$ Since $y$ is in $\bar{X}$, we can find an $x$ in $X$ such that $$|x-y|\leq \epsilon /2$$Now how do I put this together with the triangle inequality to draw my conclusion?