Question

Prove that if the sequence (an) is convergent with the limit 0, and the sequence (bn) is bounded, then the sequence (an bn ) is convergent with limit 0

Answer 1

If {bn} is bounded there exists such M > 0 that |bn| < M at any n. Thus
|an bn| = |an| |bn| < M |an| , n=1,2,....
The infinitesimal& sequence multiplied by some constant is also convergent with limit 0.