 Hadamard remainder near a compact set

For a Hadamard differentiable functional with continuous linear derivative $L$, define $R_t(h)=(\Phi (s+th)-\Phi (s))/t-L(h)$. For every compact $K$ and positive $ϵ$, some positive $\delta ,a$ make $|R_t(h)|<ϵ$ whenever $\mathrm{dist}(h,K)<\delta$ and $0<|t|<a$. If this failed, choose $t_j\to 0$, directions within $1/j$ of $K$ and remainders at least $ϵ$. Compactness gives a convergent subsequence of nearby points of $K$, hence convergent directions. The derivative definition and continuity of $L$ give remainder tending to zero, a contradiction. With a tight weak limit, the Portmanteau theorem makes random directions lie near a suitable compact set with arbitrarily large probability. Thus this lemma proves the functional delta method without assuming separability of the entire normed space or uniform differentiability on its whole unit ball.