Sequences and Bijections: Exploring Relationships and Implications

[dmatador]

Consider two sequences, {a_n} and {b_n}.

If there is a one-to-one correspondence between these sets, can we conclude anything about their behavior considering, say, that we know that one is convergent?

Going further, can we conclude anything about the series resulting from these sequences?

 

[gb7nash]

It can be shown that there is a one-to-one correspondence between A = {1,1/2,1/4,...} and B = {1,2,4,...}. The sequence and series of A converges. The sequence and series of B diverges. However, there exists the trivial bijection from any set to itself, in which case, the series/sequence will converge in both sets or diverge in both sets.

So unfortunately, we can't make any statements on convergence or divergence from inspection alone, only knowing that there exists a bijection from one set to another. However, there might be some special properties that you could look for that both sets have in common, but I'm not aware of any.

edit: Fixed a typo. A converges and B diverges.

 

[HallsofIvy]

In fact, given any two sequences {a_n} and {b_m}, the mapping a_i-> b_i is a one-to-one correspondence, so, no, the convergence of one does not tell you anything about the other.

 

[dmatador]

Thanks for the replies. I realized that the bijection doesn't tell you anything using a counterexample similar to that one right after I posted :(