Yes, the inequality

$\| A_1 + e_1 - A_2 - e_2 \| \leq \| A_1 - A_2 \| + \| e_1 \| + \| e_2 \|$

is indeed true. This follows from the triangle inequality in vector spaces.

Explanation:

Let's rewrite the left-hand side:

$\| A_1 + e_1 - A_2 - e_2 \| = \| (A_1 - A_2) + (e_1 - e_2) \|$

According to the triangle inequality, for any two vectors $x$ and $y$, we have:

$\| x + y \| \leq \| x \| + \| y \|$

Applying this to our expression:

$\| (A_1 - A_2) + (e_1 - e_2) \| \leq \| A_1 - A_2 \| + \| e_1 - e_2 \|$

Now, apply the triangle inequality again to the vector $e_1 - e_2$:

$\| e_1 - e_2 \| \leq \| e_1 \| + \| e_2 \|$

Combining these results gives:

$\| A_1 + e_1 - A_2 - e_2 \| \leq \| A_1 - A_2 \| + \| e_1 - e_2 \| \leq \| A_1 - A_2 \| + \| e_1 \| + \| e_2 \|$

Thus, the inequality is valid.

Conclusion:

The inequality $\| A_1 + e_1 - A_2 - e_2 \| \leq \| A_1 - A_2 \| + \| e_1 \| + \| e_2 \|$ is true and follows from the application of the triangle inequality twice.

Would you like more details or have any further questions?

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Follow-Up Questions:

1. How can the triangle inequality be applied to different types of norms?
2. What is the significance of this inequality in practical applications?
3. Can you provide an example with specific vectors to demonstrate this inequality?
4. How does the triangle inequality relate to distances in geometry?
5. Are there cases where equality holds in this inequality?
6. How do norms behave under linear transformations?
7. What are some common types of norms used in mathematics?
8. How does this inequality change if $A_1$, $A_2$, $e_1$, and $e_2$ are complex vectors?

Tip:

In mathematical proofs, the triangle inequality is a powerful tool that can simplify the analysis of vector norms and distances.