Final answer:
In mathematics, we can prove that if two sequences an and bn are convergent, then the limit of their product anbn is equal to the limit of an multiplied by the limit of bn. This proof involves assuming that the limits of an and bn exist and are finite numbers, and then using the properties of limits and algebraic manipulation to show that the limit of anbn is equal to the product of the limits of an and bn.
Step-by-step explanation:
Proof:
We want to prove the statement: If an and bn are convergent, lim (anbn) = lim(an) x lim(bn).
Let's assume that lim(an) = a and lim(bn) = b, where a and b are finite numbers.
Since an and bn are convergent, we can define two sequences {cn} and {dn} such that cn = an - a and dn = bn - b.
Using the properties of limits, we have:
- lim(cn) = lim(an - a) = lim(an) - a = a - a = 0
- lim(dn) = lim(bn - b) = lim(bn) - b = b - b = 0
Now, let's express anbn as (a + cn)(b + dn) and expand using the distributive property:
anbn = (a + cn)(b + dn) = ab + adn + bcn + cndn = ab + adn + bcn + 0 = ab + adn + bcn.
Since lim(cn) = lim(dn) = 0, we can take the limit as n approaches infinity:
lim(anbn) = lim(ab + adn + bcn) = ab + a x 0 + b x 0 = ab.
Therefore, we have proven that lim(anbn) = lim(an) x lim(bn).