Image of a compact set under a continuous function

Image of compact. Let $K$ compact and $f$ continuous. Fix a sequence $y_n$ in $f(K)$. There is $x_n \in K$ such that $y_n = (f(x_n))$. By compacity of $K$, we can extract a subsequence $(x_{n_k}) \subset (x_n)$ such that $x_{n_k} \to x\in K$. Since $f$ is continuous, we have $y_{n_k} = f(x_{n_k})\to f(x)\in f(K)$. Any sequence of $f(K)$ has a converging subsequence, hence $f(K)$ is compact.