Problem 4 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

Prove that ||\(x|-| y|| \leq|x-y|\).

Step-by-Step Solution

Answer

By using the triangle inequality, we proved that \[ ||x| - |y|| \leq |x - y| \].

Step 1: Understand the inequality

The given inequality is \[||x| - |y|| \leq |x - y|\].This means the absolute difference of the absolute values of x and y should be less than or equal to the absolute value of their difference.

Step 2: Consider the Triangle Inequality

Recall that the triangle inequality states:\[ |a + b| \leq |a| + |b| \].We will use this to prove our given inequality.

Step 3: Apply the Triangle Inequality

Let \[ a = x - y \] and \[ b = y \].Then by the triangle inequality:\[ |(x - y) + y| \leq |x - y| + |y| \].This simplifies to:\[ |x| \leq |x - y| + |y| \].

Step 4: Apply the Triangle Inequality Again

Similarly, let \[ a = y - x \] and \[ b = x \].Again, by the triangle inequality:\[ |(y - x) + x| \leq |y - x| + |x| \].This simplifies to:\[ |y| \leq |x - y| + |x| \].

Step 5: Combine Results

We have \[ |x| \leq |x - y| + |y| \] and \[ |y| \leq |x - y| + |x| \].Subtract \[ |y| \] from both sides of the first equation:\[ |x| - |y| \leq |x - y| \].Taking absolute values, we get:\[ ||x| - |y|| \leq |x - y| \].

Key Concepts

Absolute Value

Absolute value measures the 'distance' a number is from zero on the number line. It's always non-negative, and it disregards the number's sign. For any real number \(a\), the absolute value is written as \(|a|\).

• If \(a \geq 0\), then \(|a| = a|\).
• If \(a < 0\), then \(|a| = -a\).

For example, \(|3| = 3\) and \(|-3| = 3\). This concept is fundamental for understanding inequalities involving absolute values, such as our given inequality \[||x| - |y|| \leq |x - y|\].

Triangle Inequality

The triangle inequality is a crucial principle in mathematics, particularly useful in proofs involving absolute values. It states that for any real numbers \(a\) and \(b\):\[|a + b| \leq |a| + |b|\]This means the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values. This property gets its name because it mirrors a fundamental property of triangles: the length of any side of a triangle cannot exceed the sum of the lengths of the other two sides.To demonstrate with an example: if \(a = 2\) and \(b = -5\), then:\[|a + b| = |2 - 5| = |-3| = 3\]And\[|a| + |b| = |2| + |-5| = 2 + 5 = 7\]Clearly, \(|2 - 5| \leq |2| + |-5|\) or \(3 \leq 7\), satisfying the triangle inequality.

Inequality Proofs

Proofs involving inequalities like the triangle inequality require methodical steps. Let's break down what you saw in the solution.Step 1: Understanding the InequalityWe started with \[||x| - |y|| \leq |x - y|\]. This requires grasping absolute values and their properties.Step 2: Using Triangle InequalityWe applied the triangle inequality by choosing suitable values for \(a\) and \(b\). Let \([a = x - y]\) and \(b = y\), and then simplified further.Step 3: Repeating with ReverseSimilarly, we swapped values to \(a = y - x\) and \(b = x\) to get a similar form. This helps solidify the understanding that both forms hold true.Step 4: Combining ResultsFrom the simplified results, we subtracted and took absolute values to reach our final expression, proving that \[||x| - |y|| \leq |x - y|\].Breaking it into these steps clarifies each transformation and ensures a logical flow.

Mathematical Proofs

Proofs in mathematics serve to show the truth of a statement using a series of logical steps. They often involve established theorems and properties, like the triangle inequality.When constructing a proof:

• Identify the theorem or rule relevant to the problem.
• Break the problem into smaller, manageable steps.
• Apply the theorem step-by-step.
• Ensure each step logically follows from the previous one.
• Conclude by combining intermediate results to arrive at the final proof.

In our case, we utilized the triangle inequality repeatedly to prove the given statement about absolute values. Each step was carefully chosen to show how one part of the inequality leads to the final form, reinforcing the logical sequence fundamental to mathematical proofs.