How to Prove the Inequality for a Function: sup |f| – inf |f| ≤ sup f – inf f

In this article, we will discuss how to prove the inequality: sup |f| – inf |f| ≤ sup f – inf f, for a given function f. This inequality relates the supremum and infimum of the absolute value of a function to the supremum and infimum of the function itself. Let’s break down the steps and the reasoning involved in proving this inequality.

Understanding the Terms: Supremum and Infimum

Before diving into the proof, let’s first recall what supremum (sup) and infimum (inf) are. The supremum of a set is the least upper bound, while the infimum is the greatest lower bound. For a function f, the supremum and infimum are the highest and lowest values the function can attain, respectively, over its domain.

In the case of the absolute value function |f|, we are considering the supremum and infimum of the values of |f(x)|, not the values of f(x) itself. This leads to the inequality we are trying to prove.

Breaking Down the Inequality

We are tasked with showing that: sup |f| – inf |f| ≤ sup f – inf f. The left-hand side represents the difference between the supremum and infimum of the absolute value of the function f, while the right-hand side is the difference between the supremum and infimum of f itself.

Since the absolute value function |f| always gives non-negative values, we know that sup |f| ≥ sup f and inf |f| ≥ inf f. This basic property helps us understand the relationship between the left-hand and right-hand sides of the inequality.

Steps to Prove the Inequality

To prove the inequality, we can follow these steps:

• Step 1: Recall the definitions of supremum and infimum.
• Step 2: Use the property that |f(x)| is greater than or equal to f(x) for all x, which implies that sup |f| ≥ sup f and inf |f| ≥ inf f.
• Step 3: Show that the difference between the supremum and infimum of |f| is less than or equal to the difference between the supremum and infimum of f, using the relationships described above.

By following these steps, we can prove the inequality.

Example and Application

Let’s consider a simple example where f(x) = x for x in the interval [−2, 2]. The supremum and infimum of f(x) are:

• sup f = 2, inf f = −2.

The absolute value function |f(x)| = |x| has the following supremum and infimum:

• sup |f| = 2, inf |f| = 0.

Now, if we compute the differences:

• sup |f| – inf |f| = 2 – 0 = 2.
• sup f – inf f = 2 – (−2) = 4.

Clearly, we see that 2 ≤ 4, which satisfies the inequality. This is just one example of how the inequality holds in practice.

Conclusion

In conclusion, the inequality sup |f| – inf |f| ≤ sup f – inf f holds true by using the basic properties of supremum and infimum. By understanding these definitions and their relationships, we can easily prove this inequality and apply it to various functions. This inequality can be very useful in mathematical analysis, particularly when dealing with functions that involve absolute values.