Here are two basic questions to which I currently do not know the answer:
(1) If $a > 0$ and $b$ are integers (where $b$ is negative), then if there exists a (necessarily) negative integer $c$ such that $b = ac$, then is it true that $a \leq |b|$?
(2) Does the inequality $|c - d| \leq |c| + |d|$ hold for all integers $c$ and $d$?
MY ATTEMPT
Trying special cases for (1): Let $b = -36$, $a = 4$, and $c = -9$. Then $b=ac$, and the inequality $a \leq |b|$ holds. (How about if $c = -1$? Then we have $b = -a$, and it is then true that $a \leq |b| = |-a| = a$.) However, I currently have trouble articulating a proof for (1).
Trying special cases for (2): Let $c = -d$. Suppose that $c > 0$. Then $$2c = |2c| = |c - d| \leq |c| + |d| = |c| + |-c| = c + c = 2c,$$ which does hold. (Again, I currently have trouble articulating a proof for (2), as I have doubts if it is valid for all integers $c$ and $d$.)
(1) follows since $|b|=a|c|\geq a$ (since $|c|\geq 1$)
(2) is true by the triangle inequality (holds for real numbers $c,d$ more generally)