3rd edition. Solution to example 2.27

"An = f(n) then sequence (An) converges to L <=> f(x) converges to L as x -> infinity"

I think the statement is wrong in that (An) converges to L do not => f(x) converges to L as x -> infinity.

Suppose the statement is true.

Counter example :

Let f(x) be a function s.t. f(x) = M for all x that are no natural numbers. Where M not equal L. And f(x) = Ax for x that are natural numbers. So this function satisfies An = f(n) for all natural numbers n but has the value M for all inputs that are non-natural numbers.

Although (An) converges to L, this function f do not as x -> infinity.