Final answer:
To prove the sum of limits for convergent sequences, the commutative property of addition and the properties of limits are used to show that the sum of two convergent sequences converges to the sum of their respective limits.
Step-by-step explanation:
To prove that if sequences an and bn are convergent, then the limit of their sum is the sum of their limits, we state:
lim (aₙ + bₙ) = lim(aₙ) + lim(bₙ)
We assume that the sequences an and bn converge to L and M respectively, as n approaches infinity. That is, lim(aₙ) = L and lim(bₙ) = M.
Given the properties of limits and the commutative property of addition, which states A + B = B + A, we can see that the behavior of sequences as they approach their limits allows for the combination of their limits.
For any ε > 0, there exist natural numbers N1 and N2 such that for all n ≥ N1, |an - L| < ε/2 and for all n ≥ N2, |bₙ - M| < ε/2. Thus, for all n ≥ max(N1, N2), |(aₙ + bₙ) - (L + M)| ≤ |an - L| + |bn - M| < ε
Therefore, it is proven that lim (aₙ + bₙ) = lim(aₙ) + lim(bₙ), showcasing that the sequence formed by adding two convergent sequences is itself convergent, and its limit is the sum of the individual limits.