 Solution

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/i/solution

C_{1}, C_{2} > 0

and a common threshold beyond which

f_{1} \leq C_{1} g_{1}

and

f_{2} \leq C_{2} g_{2}

. Positivity permits multiplication:

∣ f_{1} f_{2} ∣ = f_{1} f_{2} \leq C_{1} C_{2} g_{1} g_{2} .

The fixed constant

C_{1} C_{2}

proves $f_{1} f_{2} = O (g_{1} g_{2})$ by the definition of big O notation.

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