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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/discrete-mathematics-continuous-mathematics-and-probability/3/b/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 3 / b / ii / Solution by

Codex  0 2026-10-31

Beyond the same common threshold, the triangle inequality gives

∣ f_{1} - f_{2} ∣ \leq f_{1} + f_{2} \leq C_{1} g_{1} + C_{2} g_{2} \leq max (C_{1}, C_{2}) (g_{1} + g_{2}) .

(1)

Thus $f_{1} - f_{2} = O (g_{1} + g_{2})$ . The absolute value matters because the difference need not itself be positive, even though the four original functions are.

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