14. Let {fn} be a sequence on C(X) where X C R is compact. Assume that fn (x) > fn+1(x) forall x E X. Show that there is a natural number N and a bounded and continuous function f:X -R such that if n > N, thensup|fn (x) – f(x)| < 1

Given:A sequence of functions in the space of complex-valued continuous functions , where is a…

Answered: 14. Let {f} be a sequence on C(X) where…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Use the fact ”if a function f is continuous on R and (xn) a sequence in A such that, xn → x, then f(xn) → f(x)”, to prove that function g defined by g(x) =(1 if x < 0, 2 if x = 0, 3 if x > 0. is not continuous at x = 0.
2. Let x₁>1 and Xn+1 = 2 - 1/xn for n E N. show that (xn) is bounded and monotone. Find the limit. 3 Let x. >2 and r. for n N Show that (x..) is decreasing and bounded
2. Let ffng be a sequence of nonnegative measurable functions and fn(x)!f(x)onasetAof nitemeasure and gn(x)=min(g(x);fn(x)), where g is a bounded measurable which vanishes outside a set A. Moreover, g f:
(a) Suppose that f : X → Y is a continuous function and {xn} is a Cauchy sequence in X. Does { f(xn)} have to be Cauchy as well?
10. Consider f: (0, 2)→ R defined by f(x) = x". Assume that f has a limit at 0 and find that limit. (Hint: Choose a sequence (x)=1 converging to 0 such that the limit of the sequence {f(x)} is easy to determine.) 1
QUESTION 3 (2x1+4x2 = b1 x1+ cX2 = b2 %3D Suppose that the system is consistent for all possible values of b, and ba. What can be said about the coefficient c? %3D OC=2 O C 2 O It can take only one value. O It cannot be determined.
B. i. Prove directly from the definitions that if g: R → [1, ∞) is a function so that limx-c g(x) = L, then limx-c 1/g(x) = 1/L. ii. Prove the same result of the previous part, using Relating Sequences to Functions.
Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn(x) (f( х +п -1 + f(- „x + 1 + f(= + )) n is uniformly convergent on [0, 1]. What is its limit function?
Prove that the function |2x - f(x) = 32x3 6-4x does not tend to a limit as x tends to 3. Use the e- definition of continuity to prove that the function f(x) = x3 + x²-4x is continuous at the point c = 2. Prove that the function f(x) = x-2 2x+1 is uniformly continuous on the interval [1, 3], referring to any results from the module that you use.
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1. Consider a sequence of functions defined on R by + n n f(x): = (i) Sketch the graphs of f1, f2, f3 and f₁ on the same set of axes.
Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)- f(xo)-f"(x0)(x-x0)| < |x- xo|/n whenever r – xol < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)

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14. Let {f} be a sequence on C(X) where X CR is compact. Assume that fn (x) ≥ fn+1(x) for all x E X. Show that there is a natural number N and a bounded and continuous function f: X → R such that if n ≥ N, then sup|fn (x)-f(x)| < 1 XEX