The proof you presented is almost correct but can be improved for clarity and rigor. Here's a detailed review:

Comments on the Proof:

1. Starting Step: You begin by writing: $\|x\| = \|x - y + y\|$ which is a valid identity. However, it's not necessary to express $\|x\|$ this way explicitly. You could directly apply the triangle inequality to $\|x - y + y\|$.

2. Triangle Inequality Application: You correctly invoke the triangle inequality to conclude that: $\|x - y + y\| \leq \|x - y\| + \|y\|$ But more explanation is needed. It's better to explicitly state that this inequality follows directly from the definition of the triangle inequality: $\|a + b\| \leq \|a\| + \|b\|$ with $a = x - y$ and $b = y$.

3. Subtraction: In the next step, you correctly subtract $\|y\|$ from both sides to get: $\|x\| - \|y\| \leq \|x - y\|$ This step is clear and valid.

Suggested Improvement:

You can slightly rephrase the argument to improve its clarity and flow. Here's a slightly revised version of your proof:

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Revised Proof:

We aim to prove the reverse triangle inequality, which states that $\|x\| - \|y\| \leq \|x - y\|$.

By the definition of a norm, we have: $\|x\| = \|x - y + y\|$ Applying the triangle inequality $\|a + b\| \leq \|a\| + \|b\|$, with $a = x - y$ and $b = y$, gives: $\|x\| = \|x - y + y\| \leq \|x - y\| + \|y\|.$ Subtracting $\|y\|$ from both sides, we obtain: $\|x\| - \|y\| \leq \|x - y\|.$ Thus, the reverse triangle inequality holds: $\|x\| - \|y\| \leq \|x - y\|.$

This completes the proof.

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Summary:

• The logic of the proof is correct, but rephrasing some steps and explicitly mentioning the use of the triangle inequality enhances clarity.
• Make sure to state explicitly why each inequality follows from the definition of the triangle inequality.

Would you like more details on any part of the explanation?

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Related Questions:

1. Can you prove the strict reverse triangle inequality $\|x\| - \|y\| < \|x - y\|$ under certain conditions?
2. How does this inequality behave in finite-dimensional normed spaces compared to infinite-dimensional ones?
3. What other inequalities are consequences of the triangle inequality in normed vector spaces?
4. How can we generalize the reverse triangle inequality to more general metric spaces?
5. Does equality hold in the reverse triangle inequality for certain norms, like the Euclidean norm?

Tip: Always carefully identify where specific properties of the norm or space (such as the triangle inequality) are applied in your proofs for clarity.