Prove that negative numbers are closed under addition.

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So here's what I'm thinking, but it feels too simple...

All negative numbers can be added to one another and remain negative. Consider the integers $x$ and $y$, where $x,y<0$. Then take $x+y$,

then I want to say that having both x and y be negative, means that their sum is negative, but that seems too simple... like I'm using the fact I'm trying to prove to prove it.

How should I finish this?

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1 Answer

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Here is a more formal way to state your (correct) intuition. Let $\mathbb{R}^-$ denote the set of negative reals and let $x,y \in \mathbb{R}^-$.

Since $x,y \in \mathbb{R}^-$, we know $x,y<0$. Therefore,$$ x+y < x+0 = x < 0, $$hence $x+y \in \mathbb{R}^-$.

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