The (2), please proving by contradiction in a more easy way to understand.(ps: please dont copy the answer that already...
Please be more easy to understand,thanks! 14. Let 1gn) be a sequence of non-negative real-valued continuous functions defined on a closed interval [a, b]. Suppose that for each a E [a, b g monotonically, i.e., gn(x)0 and gn(x) 2 gn+1x)2... for all n E N (1) Prove that for each n E N there exists zn E a, b] such that n m)Mngn(): E [a,b) (3 Marks) (2) By contradiction, show limn-**o M ( n 0. (10 Marks) (3) Does...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
real analysis 1,3,8,11,12 please 4.4.3 4.4.11a Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
please help me make this into a contradiction or a direct proof please. i put the question, my answer, and the textbook i used. thank you also please write neatly proof 2.5 Prove har a Simple sraph and 13 cdges cannot be bipartite CHint ercattne gr apn in to ertex Sets and Court tne忤of edges Claim Splitting the graph into two vertex, Sets ves you a 8 Ver ices So if we Change tne书 apn and an A bipartite graph...