In each of the following exercises, verify the given statement carefully, proceeding step by step. Validate each step that involves an inequality by using some statement found in this section.
Let a, b, c, d ∈ R and consider each of the following statements. Decide which are true and which are false. Prove the true ones and give counterexamples to the false ones.
a) If a < b and c < d < 0, then ac > bd.
b) If a ≤ b and c > 1, then |a + c| ≤ |b + c|.
c) If a ≤ b and b ≤ a + c, then |a − b| ≤ c.
d) If a < b − ε for all ε > 0, then a < 0.
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