Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn} converges to the zero function uniformly on [0,1]. = 0. Define the sequence {fn} ->
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![Let h: [0,1] R be a continuous function with h(1) = 0. Define the sequence {fn}
by fn(x) = x"h(x) for x E [0,1]. Show that {fn} converges to the zero function
uniformly on [0,1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdff89607-66f5-4cb4-8e83-4a60bfa7ae5f%2Fb853bb49-72f2-4aad-9c55-020e93e61c57%2Fowlxmqi_processed.jpeg&w=3840&q=75)
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- 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 x1 > 1 and xn+1 = 2 − 1 / xn for n ≥ 2. Show that ( xn ) converges and find its limit.Let x_1 = 1/2 and, for n ≥ 1, x_(n+1) = √xn. Prove that the sequence (xn)^∞_n=1 converges and find its limit.
- Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.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 a > 0 and X1 = √a. Define the sequence Xn = √(a + Xn-1), n ≥ 1.Show that (Xn)n is convergent and determine its limit.
- Suppose that fn : [0, 1] → R is defined by fn(x) = x n. If 0 ≤ x < 1, then xn → 0 as n → ∞, while if x = 1, then x n → 1 as n → ∞. So fn → f pointwise where Although each fn is continuous on [0, 1], their pointwise limit f is not (it is discontinuous at 1). Thus, pointwise convergence does not, in general, preserve continuity.Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for k = 1 to infinity converges and find its limit.(b) Give a qualitative explanation for why the sequence gn(x) = xn is not equicontinuous on [0, 1]. Is each gn uniformly continuous on [0, 1]?
- Let (gn) be a sequence of continuous functions that convergesuniformly to g on a compact set K. If g(x) = 0 on K, show (1/gn) convergesuniformly on K to 1/g.1. Consider the sequence Xn = √n + 1 − √n, n ≥ 1. Prove that (xn)n isconvergent. Find its limit.Let Show that hn → 0 uniformly on R but that the sequence of derivatives (hn) diverges for every x ∈ R.