If the sequence an < BN < CN. Then when the limits of an CN are equal, the limits of BN are also equal to them?

If the sequence an < BN < CN. Then when the limits of an CN are equal, the limits of BN are also equal to them?

It's equal

The limit of an * CN BN * cn is 0 and 1 respectively. Is there any limit of an * CN BN * CN?

An * CN uncertainty
BN * cn is positive infinity

To prove the limit: let the sequence {an}, {BN} be convergent, an = n (bn-bn-1), and Liman = 0
Let {an}, {BN} be convergent, an = n (bn-bn-1), and prove Liman = 0

An = nbn-nbn-1, the convergence of sequence must have limit
For any given ε 1, there exists N1 such that a is a limit
Bn=A+α；
For any given ε 2, there exists N2 such that
Bn-1=A+β
Let n = max {N1, N2}
Let an = n {α + (- β)}, the sum of infinitesimals be infinitesimal
The function an is infinitesimal, Liman = 0