For every integer $y$, there is an integer $x$ such that $x + y < 0$

Question

Determine the truth value of the following statement and explain your answer.

Statement: For every integer $y$, there is an integer $x$ such that $x + y < 0$.

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I think the truth value is true but I'm not sure how to explain why since we're not allowed to use examples in our explanation. I tried converting the statement into the following formula

$$\forall y \exists x \left( x + y < 0 \right)$$

but don't quite know where to go from there.

Can someone please confirm my answer and help me explain it? I will gladly appreciate it!

Answer 1

The expression to hold is $$x+y<0$$

Now let $x=-y-k$ expression becomes $$(-y-k)+y<0 \Leftrightarrow$$ $$/\text{ Associative rule} : -y-k=-k-y/$$ $$-k-y+y<0\Leftrightarrow$$ $$-k<0\Leftrightarrow$$ $$0<k$$

Now we can select any $k>0$ and the proposition will hold.