Let \(L>0\) denote a Lipschitz constant of \(F\). We have
\[ \lVert F(x+h) - F(x)\rVert = \lVert DF(x)h + \mathcal{o}(h)\rVert \le L \lVert h\rVert. \]Dividing by \(\lVert h\rVert\) and taking the limit gives the result.
Let \(L>0\) denote a Lipschitz constant of \(F\). We have
\[ \lVert F(x+h) - F(x)\rVert = \lVert DF(x)h + \mathcal{o}(h)\rVert \le L \lVert h\rVert. \]Dividing by \(\lVert h\rVert\) and taking the limit gives the result.