Mathematics — Real Analysis · continuity-of-algebraic-operations
Problem
Let {an}n=1∞ and {bn}n=1∞ be sequences.
a) Suppose {an}n=1∞ is bounded and {bn}n=1∞ converges to 0. Show that {anbn}n=1∞ converges to 0.
b) Find an example where {an}n=1∞ is unbounded, {bn}n=1∞ converges to 0, and {anbn}n=1∞ is not convergent.
c) Find an example where {an}n=1∞ is bounded, {bn}n=1∞ converges to some x=0, and {anbn}n=1∞ is not convergent.