Theorem 0.1. Let f be a function continuous at a (the domain of f is an interval containing a). Suppose that (xn) is a convergent sequence of numbers whose limit is a. Then the sequence (f (xn)) is convergent to f (a), i.e.
n→∞lim f (xn) = f (a) = f ( lim
n→∞xn).
Proof. Since f is continuous at a, for every > 0, there is δ > 0 such that for any |x−a| < δ,
|f (x) − f (a)| < . Since xn → a as n → ∞, for the given δ, we choose N > 0 such that for any n ≥ N, |xn− a| < δ. Hence |f (xn) − f (a)| < for all n ≥ N. This implies that
limn→∞f (xn) = f (a).
Example 0.1. Compute lim
n→∞sin 1 n.
Since f (x) = sin x is a continuous function on R and 1n → 0 as n → ∞, then
n→∞lim sin 1 n = sin
n→∞lim 1 n
= sin 0 = 0.
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