(a) Carefully state the definition of uniform convergence of a sequence of functions {f} to a...

    

Transcribed Image Text:

(a) Carefully state the definition of uniform convergence of a sequence of functions {f} to a function f on a set X. (b) Consider the sequence of functions In(x)= x+n = n i. Find the pointwise limit of {gn} on R. ii. Show that {9n} does not converge uniformly on R. iii. Show that {g} does converge uniformly on [-M, M] for any M > 0. (c) Prove that hn(x) = x(1-x) converges uniformly to 0 on [0, 1]. (a) Carefully state the definition of uniform convergence of a sequence of functions {f} to a function f on a set X. (b) Consider the sequence of functions In(x)= x+n = n i. Find the pointwise limit of {gn} on R. ii. Show that {9n} does not converge uniformly on R. iii. Show that {g} does converge uniformly on [-M, M] for any M > 0. (c) Prove that hn(x) = x(1-x) converges uniformly to 0 on [0, 1].