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Exercice 4 Let {fulmer be the sequence of functions defined on [0, 1] by f,(x) = 20 x 1+20 n.x2 . 1. Prove that {f,), converge pointwise on [0, 1] to a function f. 2. Compute, for every n E N*, the integral = fu(t)dt . (Hint : we can use that the derivative of In(u(x)) is "( u(x) , u > 0). 3. Find lim and deduce that the sequence {f, ), of functions does not converge uniformly to f on [0 , 1]. 4. BONUS : Find sup [f,,(x)| and deduce from it that the sequence {f,), of functions does xE[0,1] not converge uniformly to f on [0 , 1]