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...

Can I have adetailed, step-by-step explanation for part (a)-1 of the following question?

Kindlyinclude the relevant reasoning/assumptions made during simplifications & calculations along with them.

Thank you very much!

Extracted text: 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 r E (0, 1]. (b) Let fn(x) = is not uniform. 14 „2n X E [0, 1]. Use the above theorem to show that the convergence (c) Is the converse of the above theorem necessary true? Justify your answer using the sequence fn(x) = nxe-n, x E [0, 1].