Question

If lim inferior a_{n} = a, a in R and lim n->infinity b_{n} = b > 0, show lim inferior (a_{n}b_{n}) = ab

Answer 1

If \(\lim \inf\ {{n\rightarrow\infty}}\) \((a_{n}) = a, a \in\mathbb{R}\) and \(\lim_{n \rightarrow \infty } b_{n} = b >0\), show \(\lim \inf\ {{n\rightarrow\infty}}\) \((a_{n}b_{n}) = ab\)

Trying to get used to typing all of this up my hand (and couldnt figure out how to type lim inferior), but wanted to make sure the question made sense.

Answer 2

I think this will work if you use the definition of a limit:
\[\forall \epsilon>0 \exists N \in \mathbb{N} a_{n}>b-\epsilon\]

Answer 3

Typo : \(a_{n}> a - epsilon\)

Answer 4

Im not sure how to relate that to limit inferior and superior, though. I've kind of set up the inequality definitions for inferior, superior, and regular limits, but I cant relate the two.

Answer 5

Show that \(\liminf_{n \to \infty} b_n = b\) the rest should be easy.