Math 307
Final Homework, not to be turned it
1. Let an=(โ1)n
n+1 . Prove, from the definitions, that the sequence aconverges to 0.
2. Let an=3n2
n2+1 . Prove, from the definitions, that aconverges to 3.
3. Let an=n3+ 1. Prove, from the definitions, that the sequence adoes not converge to any real number.
4. Let f:RโRbe given by f(x) = 2xโ5. Prove, from the definitions, that fis continuous at 1.
5. A function f:SโTis said to be onto if (โxโT)(โaโS)[f(a) = x].
(a) Let f:Z5โZ5be given by f(x) = x2. Is this function onto?
(b) Let f:Z7โZ7be given by f(x) = x3. Is this function onto?
(c) Suppose given that f:SโTis onto, and that AโT. Prove that f(If(A)) = A.