RCISE 3 MTH3140 only 42410 marks We want to prove the follow

RCISE 3. (MTH3140 only) (4+2+4-10 marks) We want to prove the following proposition, mentioned in the lecture s. Proposition 1 Let (fn) be defined and continuous on an interval la , and differentiable on (a, b). Let cE a, b. Assume that (fn(c)) converges and that () converges uniformly on (a, b). Then (f.) converges uniformly on [a e 6/9 Let (f.) be a sequence of functions that are continuous on a, o] and differentiable on (a, b). Use the Lipschitz estimate to prove that lfa(z) _ f,(z) _ Un (c)-(c))l Ib-al supyE(a,b) IKXy)-y)I for all z E (,b] and all n, p E N (make explicit the function on which you use the Lipschitz estimate).

Solution

1.

we are given that

fn is continuous on [a,b]

and differentiable on (a,b)

and c E (a,b)

=> that c is a root of the f

therefore f(c)=0

since x=c is a zero of the function

then at x=c the fucntion fn(c) converges

and f\'n converges uniformally on (a,b)

and since there is a sign change in the interval [a,b]

=> fn will be convergent uniformally on the interval [a,b]