In this case, yes. In general, I do think of this as an infinite sequence. There may be exceptions, however, and depends on context.

First question: Is{an}an infinite sequence? (In general, when I see this notation in analysis should I always assume an infinite sequence?)

Not in general. Consider the function if , otherwise. The infinite sequence has limit and satisfies for all . But . You need stronger conditions (i.e. function is continuous) for this statement to be true. For your example, it would be so since is analytic.Since all sequences converge to the point a, is it true thatf(an) = f(a)?