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NSWMaths Extension 1Quick questions

Proof (ME-P1)

Quick questions on Mathematical induction for inequalities: the technique and the algebraic care

11short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the structure?
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To prove E(n)≤F(n)E(n) \le F(n) (or strict, or reversed) for all n≥n0n \ge n_0:
What is the strategy in the step?
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The standard technique is to show that, going from n=kn = k to n=k+1n = k + 1, the "growth" of one side is at least as much as the "growth" of the other.
What is starting at n0≠1n_0 \neq 1?
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Many inequalities are only true for n≥n \ge some threshold. Set the base case at that threshold.
What is algebraic care?
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Inequalities require you to be careful about the direction of the inequality when you multiply by a quantity. Always check whether the multiplier is positive (preserves direction) or negative (reverses).
What is factorial versus exponential?
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Prove n!>2nn! > 2^n for n≥4n \ge 4.
What is sum inequality?
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Prove ∑i=1n1i2≤2−1n\sum_{i = 1}^{n} \frac{1}{i^2} \le 2 - \frac{1}{n} for all positive integers nn.
What is a simple double?
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Prove 2n≥n+12^n \ge n + 1 for all positive integers nn.
What is algebra error in the step?
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Multiplying both sides by something requires care about the sign. If you multiply by a negative, the inequality flips.
What is skipping the strict vs weak distinction?
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Strict inequality (>>) does not generally follow from ≥\ge on each step unless one step is strict.
What is mistaking the hypothesis?
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The hypothesis is the inequality at n=kn = k, not at n=k+1n = k + 1.
What is trying to "close" the gap by inequality manipulation alone?
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Sometimes you need to verify that the algebraic implication actually holds for k≥n0k \ge n_0. Check the threshold. :::

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