Solve the following problems for extra credit on the midterm and/or the final.

1. Prove the sequence{ n2n

}converges to 0.

2. Prove that {(1)n} does not converge.

For the following, determine the convergence or nonconvergence and prove your conclusion.

3. pn =2n+ 1

n

4. pn =1n

5. pn = sinpin

For 6, 7, and 8 give the negation of each of the following statements.

6. There exists an x such that x < 4.

7. There is a real number x such that if y is any real number, then x+ y = 2.

8. If X is a connected topological space and f : X Y is a continuous function from X to a topologicalspace Y , then the image of f , denoted f(X) is also connected.

9. A group G is cyclic provided that there is a member of a of G such that for each member g of G, there isan integer n such that an = g. Explain in a useful way what it means to say that a group G is not cyclic.