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Let $M \in \mathbb{R}$. There is no element $e$ in an empty set such that $e \gt M$. It follows that all elements in empty set must be less than $M$. Similarly, we can't find any element $e$ in an empty set such that $e < M$. Hence, $M$must be the lower bound for the empty set.

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    this is correct. And it leads to the deeply unintuitive statement that the least upper bound for the empty set is $-\infty$ and the greatest lower bound is $+\infty$. Worth noting that using this convention has the nice property that the least upper bound of $A\cup B$ is the greater of the two upper bounds, and similarly for greatest lower bound. – 2017-01-07
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    It helps if you think of element $e$ as _a_ lower bound, not _the_ lower bound. – 2017-01-07

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Mostly common is to define $$\inf \varnothing := + \infty$$ and $$\sup \varnothing := -\infty$$ which is incorporated by your (correct) proof.