Help! Proving, analysis..?

Question

Prove that if,

If |x-a| < min [ε/(2(|b|+1)), 1] and |y-b| < [ε/(2(|a|+1),

then |xy – ab| < ε

Answer 1

|xy-ab| = |(x-a)b + x(y-b)| = |(x-a)b + (x-a)(y-b) + a(y-b)|

Now use triangle inequality:

|xy-ab| &lt;= |b| |x-a| + |x-a||y-b| + |a| |y-b|

&lt; ε |b|/(2|b| + 2) + ε/(2|a| + 2) + |a| ε/(2|a|+2)

= ε |b|/(2|b| + 2) + ε/2

&lt; ε /2 + ε/2 = ε

Where we used that |b|/(|b|+1) &lt; 1.