The trivial inequality is an inequality that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness.
For all real numbers , , equality holds if and only if .
We proceed by contradiction. Suppose there exists a real such that . We can have either , , or . If , then there is a clear contradiction, as . If , then gives upon division by (which is positive), so this case also leads to a contradiction. Finally, if , then gives upon division by (which is negative), and yet again we have a contradiction.
Therefore, for all real , as claimed.
The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
Suppose that and are nonnegative reals. By the trivial inequality, we have , or . Adding to both sides, we get . Since both sides of the inequality are nonnegative, it is equivalent to , and thus we have as desired.
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