MHB Is the Absolute Convergence Theorem Sufficient to Prove This?

  • Thread starter Thread starter alexmahone
  • Start date Start date
  • Tags Tags
    Convergence
alexmahone
Messages
303
Reaction score
0
Prove: if $\sum a_n$ is absolutely convergent and $\{b_n\}$ is bounded, then $\sum a_nb_n$ is convergent.

My working:

$|b_n|\le B$ for some $B\ge 0$.

$|a_n||b_n|<B|a_n|$

Since $\sum|a_n|$ converges, $\sum|a_n||b_n|=\sum|a_nb_n|$ converges.

So, $\sum a_nb_n$ converges. (Absolute convergence theorem)

Is that okay?
 
Last edited:
Physics news on Phys.org
Yes it's OK. Just one thing we have in general $|a_n|\cdot |b_n|\leq |a_n|\cdot B$ in general (this inequality is not strict in general).
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
Back
Top