5. Let f,lr)- x *a. Show that {h} converges uniformly to 0 on [0, a] for...
- Let fm (x)= 7* (0<x< 1). Show that { {m} -, converges pointwise on [0, 1]. If f(x)= lim fn(x) (0<x< 1), is there an N EI such that In(x)-f(x)}< (n>N) for all x € [0, 1] simultaneously?
a < 1. Show the series on -a, a] to onverges uniformly 25.9 (a) Let 0 (b) Does the series Explain converge uniformly on (-1,1) to =0
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
5. Let fn(x) = x"/n on [0, 1]. Show that (fr)nen converges uniformly to a differentiable function on [0, 1], but (f%) does not converge uniformly neN on [0, 1].
Real Analysis question, give clear writing please Let h(x) be the function on (0, 1) defined by ſi x <1 h(x) = 2 X=1 (a) For any P, what is the value of L(f,P)? (b) Can you find a P such that U(f,P) is within 1/10 of L(f,P)? (c) Show that h is integrable.
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
5. Let X be uniformly distributed in [0, 1]. Given X = x, the r.v. Y is uniformly distributed in 0, x for 0<x<1 (a) Specify the joint pdf fxy(x,y) and sketch its region of support Ω XY. (b) Determine fxly(x1025). (c) Determine the probability P(X〈2Y). (d) Determine the probability P(X +Y 1)
5. Let f(x) = cosx where 0<x< . Find the optimal step size h if the C.D.F of order 0(h) is used to estimate f'(xo).
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.