Question 1118504
<br>
x and y are both positive and less than 1; that means the product xy is less than x and less than y.<br>
So the expression x + y - xy is always positive; i.e., the inequality <br>
{{{0 < x + y - xy}}}<br>
is always true.<br>
To prove the other inequality<br>
{{{x + y - xy < 1}}}<br>
rewrite the statement to be proved as<br>
{{{x + y - xy - 1 < 0}}}<br>
Then
{{{x(1-y)-(1-y) < 0}}}
{{{(x-1)(1-y) < 0}}}<br>
Because x and y are both between 0 and 1, one of those factors is always negative and the other is always positive, so the product is always negative.<br>
So the second inequality is also always true, making the original compound inequality always true.