Final answer:
To prove that the convergence of an infinite series implies the limit of the function terms is zero, we use the definition of convergence and the properties of a Cauchy sequence.
Step-by-step explanation:
The question concerns the convergence of an infinite series of functions and the behavior of the individual functions within the series at the limit as n approaches infinity. To prove that if an infinite series ∑n=1∞fn is convergent on a domain D, then limn→∞fn(x)=0 for each x∈D, we can use the definition of convergence for series of functions. If the series converges, then the partial sums of the series form a Cauchy sequence. As n increases, the difference between the partial sums sn = f1 + f2 + ... + fn and sn+1 = sn + fn+1 gets arbitrarily small, which implies that fn+1(x) must approach 0 as n approaches infinity for each x in D.