### Basic property of point-set distance:

**Lemma.** For $a, b \in \mathbb{R}^d$ and $S \subseteq \mathbb{R}^d$ closed, non-empty, we have:
$$
\newcommand{\dist}{\mathrm{dist}}
\dist(a, S) - \Vert a-b\Vert \leq \dist(b, S).
$$

**Proof.**
The claim is equivalent to:
$$
\dist(a,S) \leq \dist(b,S) + \Vert a - b \Vert
$$

We recall that the definition of the distance is
$$
\dist(a,S) = \inf_{z \in S} \Vert a - z \Vert.
$$
This implies, in particular, $\dist(a,S) \leq \Vert a - P_S(b) \Vert$, since $P_S(b) \in S$.
Therefore,
$$
\dist(a,S) \leq \Vert a - P_S(b) \Vert = \Vert a - b + b - P_S(b) \Vert = \Vert a - b \Vert + \Vert b - P_S(b) \Vert
$$

## Application in our setting
Hence
$$
\begin{aligned}
\dist(x_{l-1}, S_l)
&\geq \dist(x_0, S_l) - \Vert x_{l-1} - x_0 \Vert
\\
&= \dist(x_0, S_l) - \Vert -\sum_{j=1}^{l-1} v_j \Vert
\end{aligned}
$$
where we apply the lemma with $b = x_{l-1}$, $a = x_0$ and we use that $x_{l-1} - x_0 = -\sum_{j=1}^{l-1} v_j$.