13 a. Let Let (and new be a sequence of reel numbers and let o cael....
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...
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Number 6 please S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
5a. Let (and nyl be the sequence given by An= 5nt - dn t 3 No ů What is the limit of can)? (1) Prove your answer in ) using the definition of the limit of a sequence : b. het (Audna, be a sequence ce of real numbers. In each of the following cases, check if the sequence Cand is a cauchy sequence. Justify your answers is an on 5 มาก
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...
(2) (a) Let {n}nen be a sequence of complex numbers. Show that if lim, toon = 2, then lim 21+z2+ + + Zn 100 (b) Using (a), find the limit limn7 (m_ +i i zm).
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample (2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...