is it true that abs(a-b) <= abs(a) + abs(b) ?

Yes, it is true that:

\[
|a - b| \leq |a| + |b|
\]

This is known as the **Triangle Inequality**. It holds for all real numbers \(a\) and \(b\).

### Here's a brief explanation:

The Triangle Inequality expresses that the absolute difference between two numbers is always less than or equal to the sum of their absolute values. This can be understood geometrically as well: in a triangle, the length of any one side is always less than or equal to the sum of the lengths of the other two sides.

### Proof outline:
1. The inequality follows from the definition of absolute value and some basic properties of inequalities.
2. Consider the fact that for any real numbers \(a\) and \(b\), the absolute value of their sum satisfies:
   \[
   |a + b| \leq |a| + |b|
   \]
   This is also a version of the Triangle Inequality for sums.
3. By expressing \(a - b = a + (-b)\), we can apply this inequality to prove:
   \[
   |a - b| \leq |a| + |-b| = |a| + |b|
   \]
   which is exactly the result you asked about.

Would you like more details on how to formally prove this, or any specific part of this concept clarified?

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Here are 5 related questions for deeper understanding:
1. What is the geometric interpretation of the Triangle Inequality in the context of vectors?
2. How does the Triangle Inequality apply to complex numbers?
3. Can you extend the Triangle Inequality to three or more terms?
4. In what mathematical fields does the Triangle Inequality play a key role?
5. How does the Triangle Inequality help in analyzing distances in metric spaces?

**Tip:** The Triangle Inequality is widely used in distance and normed spaces, helping to establish limits and convergence.

Is it true that abs(a-b) <= abs(a) + abs(b)?

Explore the Triangle Inequality theorem, which states that the absolute difference between two numbers is always less than or equal to the sum of their absolute values. Learn how this inequality applies to real numbers, complex numbers, and metric spaces, with a proof outline and related mathematical concepts.

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