Show that if {fn} converges uniformly on {fn} converges uniformly on [a, b]. (a,b) and {fn(a)}, {fn(b)} converge, then
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![Show that if {f} converges uniformly
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?For any integer n ≥ 1 and any x ∈ (0,∞), define fn(x)= nx/(1+nx) (a) Let a > 0 be given. Prove that {fn} converges uniformly on the interval (a, ∞). (b) Prove that {fn} does not converge uniformly on (0,∞).Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].
- Let fn(x) = x/(n^2+x^2) for x ∈ R. Show that the sequence {fn} converges uniformly to the function that is everywhere zero.Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (fn ) converges uniformly on [a, b]. If thereexists a point x0 ∈ [a, b] where fn(x0) is convergent, then (fn) converges uniformly on [a, b]. Proof. Exercise 6.3.7. Combining the last two results produces a stronger version of Theorem 6.3.1.Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.
- (b) A sequence (fn) of differentiable functions such that (fn ) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?if fn converges uniformly to f and each fn is increasing then f is increasing. Determine true of False and justify your answer.
