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**Plato** Suppose that $\displaystyle \alpha = \sup (A)\,\& \,\beta = \inf (B)$.

If $\displaystyle \beta < \alpha $ then by definition of supremum and infimum $\displaystyle \beta $ is not an upper bound of A, so $\displaystyle \left( {\exists c \in A} \right)\left[ {\beta < c \le \alpha } \right]$ but c is not a lower bound of B. That means $\displaystyle \left( {\exists d \in B} \right)\left[ {\beta \le d < c} \right]$, but that means d is a upper bound of A which is impossible because $\displaystyle {d < c \le \alpha }$. That means that $\displaystyle \alpha \le \beta $.

This time suppose that $\displaystyle \alpha < \beta$ which implies $\displaystyle \alpha < \frac{{\alpha + \beta }}{2} < \beta $.

But this means that $\displaystyle \frac{{\alpha + \beta }}{2}$ is an upper bound for A but it cannot belong to B. That is a contradiction. Thus $\displaystyle \alpha = \beta $.