Walk me through this proof.
Not 'clearly' or 'trivially'. Real motivations for each step, why this proof strategy over another, and where the difficulty actually lives.
You have a proof — from a paper, a textbook, or a Wikipedia page — and the presentation is compressed to the point of unreadability. 'It follows immediately that...' is doing all the work. This prompt unpacks the proof with the motivation for each step, so you understand not just what happened but why.
THE PROMPT
Walk me through the following proof step by step. My background: [YOUR MATH BACKGROUND — e.g., "linear algebra fluent, some real analysis, no measure theory"]. Structure as: **Statement of what's being proved** Restate the theorem or claim in your own words. If the statement uses notation that would be unfamiliar to my level, translate it. **The proof strategy in one paragraph** Why does this proof work? What's the trick or the technique? Naming this up front turns the proof from a sequence of moves into a plan. **The proof, step by step** Number each step. For each step: - State the step (what claim is being made). - Justify it (why is this step true). - Motivate it (why did we take this step at this point — what would break if we didn't). **The step that carries the weight** Identify the one step in the proof that is doing the actual work. Often the proof presents everything as equal in importance; usually one step is where the difficulty lives. Name it. **What I should be careful about** 1-2 subtleties. Places where a first reading would let me think I understood without actually understanding. Common misreadings. Do not use "clearly", "trivially", "obviously", or "it is easy to see that". If a step is genuinely easy, say why in one line. Proof: [PASTE PROOF]
**Statement of what's being proved**
The claim is that any convergent sequence in a metric space is a Cauchy sequence. In plainer language: if a sequence gets arbitrarily close to some limit L, then the terms of the sequence also get arbitrarily close to each other.
**The proof strategy**
The trick is to move from "close to L" (which we're given) to "close to each other" (which we want) by inserting L as a middle term and using the triangle inequality. Whenever you see a proof that turns one closeness claim into another, look for the triangle inequality.
**The proof, step by step**
1. **Fix ε > 0.** We want to show the sequence is Cauchy, which means: for any tolerance ε we're given, we can find a threshold N past which all terms are within ε of each other. So we fix ε first and prove the existence of that N.
2. **By convergence, there exists N such that for all n ≥ N, d(x_n, L) < ε/2.** This uses the definition of convergence directly. We use ε/2 rather than ε because we're going to add two of these together and want the sum to be ε. This is the setup move.
3. **For any m, n ≥ N, we have d(x_m, x_n) ≤ d(x_m, L) + d(L, x_n).** Triangle inequality. This is the step that connects "close to L" to "close to each other".
4. **d(x_m, L) + d(L, x_n) < ε/2 + ε/2 = ε.** Substituting from step 2. Both m and n are ≥ N, so both terms are < ε/2.
5. **Therefore d(x_m, x_n) < ε for all m, n ≥ N.** Which is the definition of Cauchy.
**The step that carries the weight**
Step 3 — the triangle inequality — is the real move. Steps 1, 2, 4, and 5 are bookkeeping. Once you see step 3, the proof is essentially done.
**What I should be careful about**
- The proof does not require the metric space to be complete. Every convergent sequence is Cauchy; the converse (every Cauchy sequence converges) is what completeness gives you.
- The ε/2 trick is a common device in analysis. It's not deep — it's just "if you're going to add two things, budget half of your total for each". Recognizing this pattern will save you time in many analysis proofs.