Question

Limsup and Liminf We define the lim inf and lim sup of a sequence as follows: lim inf n?? an = lim n?? inf{ak : k ? n} lim sup n?? an = lim n?? sup{ak : k ? n}. 1. Let {an} be a bounded sequence. Show that lim inf n?? an and lim sup n?? an exist and are in R. 2. Let {an} be an unbounded sequence. Show that either lim inf n?? an = -? or lim sup n?? an = ? (or possibly both). 3. Let {an}, {bn} be two sequences. Show that lim inf n?? an + lim sup n?? bn ? lim sup n?? (an + bn) ? lim sup n?? an + lim sup n?? bn. Furthermore, find a pair of sequences for which the second inequality is strict.

          Limsup and Liminf
We define the lim inf and lim sup of a sequence as follows:
lim inf n?? an = lim n?? inf{ak : k ? n}
lim sup n?? an = lim n?? sup{ak : k ? n}.

1. Let {an} be a bounded sequence. Show that lim inf n?? an and lim sup n?? an exist and are in R.
2. Let {an} be an unbounded sequence. Show that either lim inf n?? an = -? or lim sup n?? an = ? (or possibly both).
3. Let {an}, {bn} be two sequences. Show that
lim inf n?? an + lim sup n?? bn ? lim sup n?? (an + bn) ? lim sup n?? an + lim sup n?? bn.
Furthermore, find a pair of sequences for which the second inequality is strict.

Limsup and Liminf
We define the lim inf and lim sup of a sequence as follows:
lim inf n?? an = lim n?? infak : k ? n
lim sup n?? an = lim n?? supak : k ? n.

1. Let an be a bounded sequence. Show that lim inf n?? an and lim sup n?? an exist and are in R.
2. Let an be an unbounded sequence. Show that either lim inf n?? an = -? or lim sup n?? an = ? (or possibly both).
3. Let an, bn be two sequences. Show that
lim inf n?? an + lim sup n?? bn ? lim sup n?? (an + bn) ? lim sup n?? an + lim sup n?? bn.
Furthermore, find a pair of sequences for which the second inequality is strict.

Added by Danielle F.

Question

Please give Ace some feedback