Math Help - Convergence proof

1. Convergence proof

Using the given that the sequence {1/n} converges to 0, prove that none of the following assertions are equivalent to the definition of convergence of a sequence {an} to the number a:

a. For some ε>0 there is an index N such that
|an-a|<ε for all indices n≥N.

b. For each ε>0 and each index N
|an-a|<ε for all indices n≥N.

c. There is an index N such that for every number ε > 0,
|an-a|<ε for all indices n≥N.

2. Originally Posted by uconn711
Using the given that the sequence {1/n} converges to 0, prove that none of the following assertions are equivalent to the definition of convergence of a sequence {an} to the number a:

a. For some ε>0 there is an index N such that
|an-a|<ε for all indices n≥N.
Consider the sequence

a_n=(-1)^n, n=0, 1, ..

This does not converge.

but for ε=2 |a_n-0|<ε, for all n>0.

RonL