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Problem 94: (When The Distance Between Two Numbers Is Smaller Than Every Positive Number)
The trichotemy axiom of the real numbers says that given any two real numbers $x$ and $y$, we must have exactly one of $x<y$, $x=y$, or $x>y$ be true. Let $x$ and $y$ be real numbers. Suppose that for every real number $\varepsilon>0$, we know $|x-y|<\varepsilon$. In other words, we've supposed that the distance between $x$ and $y$ is less than $\varepsilon$ for every positive real number $\varepsilon$. Use the trichotemy axiom to prove that $x=y$.
Note, we can use this fact to quickly prove that $3.9\overline{9}$ and 4 are the same real number. The fact above is also directly related to why we define limits by talking about making the distance between two things smaller than a given $\varepsilon$.
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