2 Hints

As per usual, here’s where you’ll find the problem sheet hints!

1. The ideas in this one are pretty similar to ‘Tutorial Question 1’. Here’s a potential route through this question:
   1. Since \(f\) and \(g\) are integrable, they are bounded. So, there exists a common \(M>0\) such that \(\lvert f(x) \rvert \leq M\) and \(\lvert g(x) \rvert \leq M\) for all \(x \in [a,b]\). Why does this mean that \(f\cdot g\) is bounded?
   2. Let \(P = \{x_0, \ldots x_n\}\) be a subdivision of \([a,b]\). For any interval \(I_i = [x_{i-1},x_{i}]\), use techniques/results from `Tutorial Question 1’ to show that \[\sup_{I_i}(f\cdot g) - \inf_{I_i}(f\cdot g) \leq M(\sup_{I_i}f - \inf_{I_i}f) + M(\sup_{I_i}g - \inf_{I_i}g).\]
   3. Using the above result, find a corresponding inequality relating lower and upper Riemann sums.
   4. Fix \(\epsilon >0\), and apply the Cauchy criterion to \(f\) and \(g\) separately (obtaining subdivisions \(P_1\) and \(P_2\) respectively). Using these, find a common subdivision for which \(f\) and \(g\) satisfy the Cauchy criterion, and show that with this subdivision, \(f\cdot g\) also satisfies the Cauchy criterion.
2. Firstly, why is \(f\) uniformly continuous on \([0,1]\)? Next, use the definitions for uniform continuity of \(f\) on \([0,1]\) and \(g\) on \([1,\infty)\) to find a candidate \(\delta\) for uniform continuity of \(h\) on \([0,\infty)\). Finally, with this \(\delta\), show that \(h\) satisfies the definition of uniform continuity (you’ll need three cases for the values of \(x,y\) in the definition).