Final answer:
Convergent sequences an and bn imply that lim(αan) will converge to α times the limit of an, as a constant multiple of a convergent sequence also converges to the product of the constant and the original sequence's limit.
Step-by-step explanation:
If an and bn are convergent, we can deduce about lim(αan) that it will also converge to α times the limit of an. This is because if a sequence an is convergent and approaches a limit L, so will a sequence αan, which will approach αL as its limit. .
This is based on the principle that a constant multiple of a convergent sequence is also convergent, assuming α is a finite constant.
Convergence of a sequence means that as n goes to infinity, the terms of the sequence get arbitrarily close to a specific value, the limit. In this case, αan gets arbitrarily close to αL, where L is the limit of an.