Answered: 5. Assume fn :D → R converges uniformly… | bartleby

Calculus For The Life Sciences

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.4: Improper Integrals

Problem 37E

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5. Assume fn :D → R converges uniformly to f and gn: D → R converges
uniformly to g. Prove that (fngn) converges uniformly to fg.

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Assume fn: D → R converges uniformly to f and 9n : D → R converges uniformly to g. Prove that (fngn) converges uniformly to fg.
(b) Give an example to show that the product (fngn) may not converge uniformly.
For each n = Z+ let fn : R → R be the function пх fn (2) = n|x| +1 (a) Prove that (fn) converges pointwise to some function f : R→ R. (b) Prove that (fn) does not converge uniformly.
Show that the sequence of functions {f.} where f,(x)=nxe is non-uniformly convergent on [0,1]. (by using the definition)
a) Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t respectively. Show that (h, tn) converges uniformly on A to ht.
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7. Let fn: R → R be a sequence of continuous functions which converges uniformly to a function f : R → R. Let (n) be a sequence of real numbers which converges to x E R. Show that fn(xn) → f(x).
Suppose (f) and g,) ave bwo sezuences of functions that converge unifomly on subseb ACR, Zs ib true thab the seguence (fn Jn) converges uniformly on A e PVove or give a Counterexample.

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