1. Let c > 0 be fixed and define the sequence {an} by settingan (an? + 3c)За,2 + с2а1 > 0,An+1for n E N.Determine all a1 for which the sequence converges and in such a casefind its limit.2. Prove that if f is a bounded function on [0,1] satisfying f(ax)bf (x) for 0 < x <- and a, b > 1, then lim f(x) = f(0).| 1 3. Suppose that f:[1, c0) → R is uniformly continuous. Prove thatthere is a positive M such that-If (x)|< M for x > 1.4. A function f: [0,1] → R satisfies f (0) < 0 and f(1) > 0, and thereexists a function g continuous on [0,1] and such that f + g is decreasing.Prove that the equation f (x) = O has a solution in the open interval (0,1).

Name: 1. Let c > 0 be fixed and define the sequence {an} by settingan (an?
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: 1. Let c > 0 be fixed and define the sequence {an} by settingan (an? + 3c)За,2 + с2а1 > 0,An+1for n E N.Determine all a1 for which the sequence converges and in such a casefind its limit.2. Prove that if f is a bounded function on [0,1] satisfying f