6. Further Pure Mathematics

Advanced Proofs — Quiz

Test your understanding of advanced proofs with 5 practice questions.

Practice Questions

Question 1

Which proof technique is most suitable for proving properties of elements in a set defined recursively (e.g., arithmetic expressions formed by a grammar)?

Question 2

In an epsilon-delta proof of $\lim_{x\to a}x^3=a^3$, one writes $|x^3-a^3|=|x-a||x^2+ax+a^2|$. Given $|x-a|<1$ implies $|x^2+ax+a^2|<3|a|+1$, which choice of $\delta$ in terms of $\epsilon$ and $a$ ensures the proof works?

Question 3

Which proof technique is most appropriate for proving the inequality $F_n<2^n$ for the Fibonacci sequence defined by $F_1=1, F_2=1, F_{n+2}=F_n+F_{n+1}$?

Question 4

In a proof by minimal counterexample (using the well-ordering principle), what is the initial assumption?

Question 5

To prove directly that for any odd integer $n$, the quantity $n^4 - 1$ is divisible by 8, which factorization is most useful?