Let $K$ be a compact interval in $\mathbb{R}$. Then every continous function $\phi :K\rightarrow \mathbb{R}^d$ is automatically bounded.
Is this a consequence of; the image of a compact is compact ? It's just when the author is using the word "automatically" I'm being confused.
Yes $\phi$ maps compact sets to compact sets, so its image is compact, and thus bounded, thus $\phi$ is bounded.