Real Analysis Just need to answer true or false. Let (xn) be a bounded sequence and let mare infinitely many terms in the sequence greater than m – €.lim sup xn. Then for every e > 0 there%3DIf ƒ : D → R and f(D) is a bounded set, then f is continuous on D.If ƒ : D → R and c E D is an isolated point, then f is continuous at c.If ƒ : D → R is continuous at c and c e D', then lim,→c f (x) = f(c).If ƒ : D → R and (xn) is a Cauchy sequence in D, then (f(xn)) converges.Suppose f: D → R is continuous. Then there exists x1 E D such that f (x1) > f(x)for all x E D.Let D CR be bounded and f : D → R be continuous. Then f(D) is bounded.If ƒ : D → R is continuous and bounded on D, then f assumes maximum and mini-mum values on D.

Name: Real Analysis Just need to answer true or false. Let (xn) be a bounded
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Real Analysis Just need to answer true or false. Let (xn) be a bounded sequence and let mare infinitely many terms in the sequence greater than m – €.lim sup xn. Then for every e > 0 there%3DIf ƒ : D → R and f(D) is a bounded set, then f is continuou

The first statement is True. The second statement is False. A bounded function is not always continuous. Consider the function f:D=−2π,2π→ℝ; fx=sinx. The set of values that f can take is fD=−1,0. Clearly, f is bounded but not continuous on D. The third statement is True. By the definition, an isolated point is an element of the domain such that “there exists no neighborhood of c such that it contains c.” So, the only c∈D such that x−c<δ is x=c. Then, 0=fx−fc<ε. So, f is continuous at every isolation point.The fourth statement is True. If c is an accumulation point, then for every x∈S open, 0<x−c<δ. Then, 0<fx−fc<ε. So, limx→cfx=fc. The fifth statement is True. A sequence is Cauchy if and only if it converges. The sixth statement is False. This is not always true, but it always if D is bounded. The seventh statement is False. For example, in the bounded set D=0,1, fx=1x for x∈D is continuous but f(D) is unbounded. However, this statement is true if D is closed – follows from the boundedness theorem. The eight statement is False. The argument is similar to that of statement seven. It is true if D is closed.