And similarly is a unbounded sequence times a convergent sequence bounded? I'm still getting familiar with the properties of sequences.
Is a bounded sequence times a unbounded sequence bounded?
2
$\begingroup$
calculus
sequences-and-series
-
0What if you multiply a sequence by the sequence $1,1,1,\ldots$? – 2017-02-09
-
3What simple examples of sequences do you know? Try them out. – 2017-02-09
2 Answers
1
An unbounded sequence times a sequence that has at least one nonzero convergence point is actually always unbounded.
That should answer both your questions. For a concrete counterexample, take any unbounded sequence and the bounded sequence $1,1,1,\dots$
1
A counter example shows both: $a_n = 2$ is bounded, and $b_n = n$ is unbounded.But $a_nb_n = 2n$ which is unbounded, so we don't have the first. Also $a_n = 2$ converges to $2$, so it covers your second part as well.