Final answer:
The proof establishes that the limit of a constant times a convergent sequence equals the constant times the limit of the sequence, relying on the definition of convergence and the properties of limits.
Step-by-step explanation:
The question asks to prove that, given the sequences an and bn are convergent, the limit of the sequence αan equals α times the limit of the sequence aₙ. In math notation, this is to show that lim(α aₙ) = α lim(aₙ).
To prove this, we first recall that a sequence an is convergent if it approaches some real number L as n goes to infinity. In proper notation, this is expressed as lim(aₙ) = L. By the definition of limit, for every ε > 0 there exists an N such that for all n > N, |aₙ - L| < ε.
Now, consider the sequence αaₙ. To find its limit, we multiply the above inequality by α, resulting in |α(aₙ - L)| < αε. Since the sequence an converges to L, it follows that as n approaches infinity, αaₙ approaches αL. Hence, we conclude that lim(αaₙ) = α lim(aₙ).