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- Let (fn) be the sequence defined by fn = on R. (a) Show that (fn) converges point wise. Find its limit. (b) Show that the above convergence is not uniform. (c) Show that on any bounded interval the above convergence is uniform.If the sequence of functions {f} converges in measure to two functions f(x) and g(x), then these limit functions are equivalent.Let (rn) be a sequence of real numbers. Show that the convergence of (sn) implies theconvergence of (|sn|).
- Let 00 Σ f(x) n 1 Find the intervals of convergence for f. (Enter your answers using interval notation.) Find the intervals of convergence for f' Find the intervals of convergence for f"Assume (fn) and (gn) are uniformly convergent sequences of functions. (b) Give an example to show that the product (fngn) may not converge uniformlyIf x1, y1, are two positive unequal numbers and xn = (xn-1+ yn-1)/2 and yn= sqrt(xn-1yn-1) for all n>=2, Prove that the sequences <xn> and <yn> are monotonic and they converge to the same limit. *Prove each step