A $k$-simplex is a convex hull of $k+1$ points (which are called vertices) in general position in $\mathbb{R}^n$ (for $k\le n$). A face of a simplex is a simplex spanned by a subset of its vertex set. And a collection $C$ of simplices in $\mathbb{R}^n$ is called a simplicial complex if every two simplices in $C$ meet, if at all, in a face common to both. The union of all simplices in $C$ is denoted by $||C||$. A triangulation of a topological space $X$ is a simplicial complex $C$ such that $||C||$ is homeomorphic to $X$.
Then for a triangulation $C$ of a $k$-sphere $S^k$, I am wondering is there a $\ell$-simplex in $C$ with $\ell>k$? My intuition is that it's impossible. But I don't know how to prove it.