Let {fn}1 be a sequence of continous functions on an interval [a, b]. If {fn}-1 converges uniformly to a function f on [a, b], then f is continous on [a, b]. (a) (i) Discuss the convergence of the sequence of functions fn, when fn(x) 1+ xn' where r E [0, 1]. (ii) Show that the sequence {fn}, where fn(x) = 1 is not uniformly convergent 1+ xn on x € [0, 1]. (b) Let fn(x) = is not uniform. (c) Is the converse of the above theorem necessary true? Justify your answer using the sequence fn(x)= nxe¬nª, x E [0, 1]. lt n2n, X E [0, 1]. Use the above theorem to show that the convergence

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