Let $P:H \to K$ be a nonlinear map, where $K$ is a closed convex subset of $H$ which is a Hilbert space. We know that $P$ is Lipschitz with a Lipschitz constant one.
Let $S \subset H$ be a set. We know that for every $s \in S$,
$$P(s+th) -Ps = th$$ holds, for some fixed $h \in H$, as long as $t$ is small enough (it depends on $s$).
Let $\bar S$ denote the closure of $S$. Then apparently, due to density and Lipschitzness of $P$, we have that for every $s \in \bar S$, $$P(s+th) -Ps = th + o(t)$$ where $t^{-1}o(t) \to 0$ as $t \to 0$.
Could somebody tell me how this is gotten, and how to describe $o(t)$ precisely?
This is Lemma 1 on page 620 of this paper https://projecteuclid.org/download/pdf_1/euclid.jmsj/1240432858.
You are reading the lemma wrong. The closure is not for $S$ but for the set of directions $h\in H$.
Given $\varepsilon>0$ for $h\in\overline{H}$ there exists $h_{\varepsilon}\in H$ such that $\Vert h-h_{\varepsilon}\Vert\leq\varepsilon$. Then \begin{align*} \Vert P(s+th)-P(s)-th\Vert & \leq\Vert P(s+th)-P(s+th_{\varepsilon})\Vert\\ & +\Vert P(s+th_{\varepsilon})-P(s)-th_{\varepsilon}\Vert+\Vert th_{\varepsilon}-th\Vert\\ & \leq\text{Lip }P\Vert s+th-(s+th_{\varepsilon})\Vert+0+|t|\Vert h_{\varepsilon}-h\Vert\\ & \leq\text{Lip }P|t|\Vert h-h_{\varepsilon}\Vert+0+|t|\Vert h_{\varepsilon }-h\Vert\leq(\text{Lip }P+1)\varepsilon|t|, \end{align*} and so $$ \left\Vert \frac{P(s+th)-P(s)}{t}-h\right\Vert \leq(\text{Lip }P+1)\varepsilon , $$ which shows that $$ \lim_{t\rightarrow0}\frac{P(s+th)-P(s)}{t}-h=0. $$