Question 6.T.15: If f is a continuous function on the compact set D ⊆ R, then...

Let (y_{n}) be any sequence in f(D). f(D) is compact if we can find a convergent subsequence of (y_{n}) whose limit lies in f(D). For every n ∈ \mathbb{N} there is an x_{n} ∈ D such that y_{n} = f(x_{n}). Since (x_{n}) is a sequence in the compact set D, it has, according to Theorem 6.13, a subsequence (x_{n_{k}}) which converges to some point x ∈ D. Now the continuity of f implies

y_{n_{k}} = f(x_{n_{k}}) → f(x) ∈ f(D).