5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
6. (10 points) Let fn: R+R be defined as fn(x) = (sin(ru) 2 Show fn converges pointwise but not uniformly on R.
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
5a) (5 pts) Find lim inf (xn) and lim sup (rn), for rn = 4 + (-1)" (1 - 2). Justify your answer 5b) (5 pts) Find a sequence r, with lim sup (xn) = 3 and lim inf (x,) = -2. 5c) (10 pts) Let {x,} be a bounded sequence of real numbers with lim inf (x,) = x and lim sup (x,) = y where , yER. Show that {xn} has subsequences {an} and {bn}, such that an...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim || |, ? 7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim ||...