Let $\Sigma\subset \mathbb R^n$ be a $k$-dimensional complete minimal submanifold. Let $d(x,y)$ be the intrinsic distance between $x,y\in \Sigma$ and $|x-y|$ be the Euclidean distance. Let $y\in\Sigma$ be fixed. Do we have $$|x-y|\le C\Longrightarrow \text{ there is }R>0 \text{ such that }d(x,y)\le R?$$
This is not true for general submanifolds in $\mathbb R^n$ since one can have the topologist sine curve.