Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product. That is, if $0< x< 1$ and $0< y< 1$ then prove $$x+y< 1+xy$$
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Pure inequalities such as this one are often used in the analysis of far more difficult mathematics problems: whilst the inequalities might be simple to prove in themselves, they can be surprisingly useful as tools.