We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{ s,p} = F^{ s}_{ p,p}$, and $H^{ 1,p} = F^{ 1}_{ p,2}$. When $s\to 1$, the $F^{ s}_{ p,p}$ norm becomes the $F^{ 1}_{ p,p}$ norm but BBM showed that the $W^{ s,p}$ norm becomes the $H^{ 1,p} = F^{ 1}_{ p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{ s,p}$ into $F^{ s}_{ p,q}$ for $q \in \{ p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{ R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown.
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