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3. Construct a sequence of integrable functions {fk} on [0, 2π] such that lim k→∞ 1 2π Z 2π 0 |fk(θ)| 2 dθ = 0 but limk→∞ fk(θ) fails to exist for any θ. [Hint: Choose a sequence of intervals Ik ⊂ [0, 2π] whose lengths tend to 0, and so that each point belongs to infinitely many of them; then let fk = χIk .] 5. Let f(θ) = ( 0 for θ = 0 log(1/θ) for 0 < θ ≤ 2π, and define a sequence of functions in R by fn(θ) = ( 0 for 0 ≤ θ ≤ 1/n f(θ) for 1/n < θ ≤ 2π. Prove that {fn}∞ n=1 is a Cauchy sequence in R. However, f does not belong to R. [Hint: Show that R b a (log θ) 2 dθ → 0 if 0 < a < b and b → 0, by using the fact that the derivative of θ(log θ) 2 − 2θ log θ + 2θ is equal to (log θ) 2 .]

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