Prov that $|xy-ab|-c^2< (|a|+|b|)c)$

Question

Suppose that $x,a,y,b,c \in \mathbb{R}$ such that $|x-a|<c,|y-b|<c$. Prov that $|xy-ab|-c^2< (|a|+|b|)c)$

How to prove this using field and order axioms ...please some can some help please

Answer 1

Observer that the conditions imply $c>0$ and then note that $$ |y|<c+|b|$$ and hence $$ |xy-ab|=|(x-a)y +a(y-b)|\le c|y|+c|a|<c^2+c|b|+c|a|$$