Use the Triangle Inequality, and other properties of modulus, to show that:

Question

(a) If $|a|\le 1$ then $|3+a^3|\le 4$.

(b) If $|a|\le 1/2$ and $|b|\le 1/4$ then $|4a^4+2b^2|\le 3/8$.

Answer 1

Note that $$|a^3+3| \le |a^3|+3 = |a|^3 +3$$

If we have $|a|\le 1$, then $|a|^3 \le 1$. Now you should be able to make conclusion for the first part.