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> Prove formally that $a \le x \le a \Rightarrow x = a$ for $x, a \in \mathbb R$ not using contradiction.
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I've been thinking how to prove the above statement not using contradiction.

My idea for a proof by contradiction is as follows: Suppose for contradiction that $x \neq a$ then the equality fails so $x = a$.

Suppose $a,x$ are integers or rational numbers, then proving the above statement for real numbers prove the statement for these classes of numbers also, since we can view them as real numbers ? Or should one be careful, since the ordering is defined different ?