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Let $S$ be a nonempty closed subset of a Hilbert space $X$, let $x\in X$ and $s \in S$ proving that

if $s \in proj_S(x)$ then $\forall t\in ]0,1[$, $proj_S(s+t(x-s))$ is a singleton $\{s\}$

Thank you very much

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Denote the projection onto $S$ by $P$. Then $P(x) = s =P(s)$ and for all $t$ we thus get
$$ P(s + t(x-s)) = s + tP(x-s) = s. $$

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    thank you for your answer, but i did not understande how the equality coms from ! did you consider the projection operator linear ? – 2017-02-04
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    Projections are linear, yes. To be precise: A projection on a Hilbert space $H$ is a bounded operator $P : H \to H$ such that $P^2 = P = P^*$. – 2017-02-04