Puzzle 1

Let set $\mathcal P$ be defined by a ray originated in point $O$ and passing through point $I$, and $\mathcal Q$ be some bounded convex set (shown as an ellipse here for sake of programming easiness). Let $p_0 \in \mathcal P$, and define iteratively $\{p_n\}, \{q_n\}, \ n \in \mathbb{N} $ as:

$$p_n \to q_{n+1} = \Pr_{\mathcal Q} p_n,$$ $$ q_n \to p_n : p_n\in\mathcal{P}, (p_n-q_n, q_n) = 0,$$ that is the next $q$ point is obtained by projection the last $p$ point on $\mathcal Q,$ and the next $p$ point is obtained by intersection of the perpendicular to line $O q$ in point $q$ and line $O I$.

Prove that

1. $\exists\ \hat q = \lim_{n\to\inf} q_n$ and
2. If $\mathcal P \cap \mathcal Q \neq \varnothing$, then $\hat q \in \mathcal P$; otherwise, line $O \hat q$ is tangent to $\mathcal Q$.

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