Proof that generalized directional derivative is upper semicontinuous In "Nonsmooth Optimization" by Mäkela and Neittaanmäki the definition of the generalized directional derivative is given as follows:
Definition 3.1.1 (Clarke). Let $f: \mathbf{R}^{n} \rightarrow \mathbf{R}$ be locally Lipschitz at a point $x \in \mathrm{R}^{n} .$ The generalized directional derivative of $f$ at $x$ in the direction of $v \in \mathrm{R}^{n}$ is defined by
$$
f^{\circ}(x ; v)=\limsup _{y \rightarrow x, t \rightarrow 0} \frac{f(y+t v)-f(y)}{t} \, .
$$
Then in Theorem 3.1.2. on page 30 they proof that $f^{\circ}(x ; v)$ is upper semicontinuous as a function of $(x ; v)$.
The proof starts with:
Let $\left(x_{i}\right),\left(v_{i}\right) \subset \mathbf{R}^{n}$ be sequences such that $x_{i} \rightarrow x$ and $v_{i} \rightarrow v$.
By definition of upper limit, there exist sequences $\left(y_{i}\right) \subset \mathbf{R}^{n}$ and $\left(t_{i}\right) \subset \mathrm{R}$ such that $t_{i}>0$
\begin{equation}
\left\|y_{i}-x_{i}\right\|+t_{i}< \frac{1}{i} 
\end{equation}
and
\begin{equation}
f^{\circ}\left(x_{i} ; v_{i}\right) - \frac{1}{i}\leq 
\frac{f\left(y_{i}+t_{i} v_{i}\right)-f\left(y_{i}\right)}{t_{i}} 
\end{equation}
for all $i \in \mathrm{N}$.
Why do such sequences $\left(y_{i}\right) \subset \mathbf{R}^{n}$ and $\left(t_{i}\right) \subset \mathrm{R}$ exist?
I mean by the above definition of the generalized directional derivative we have
\begin{align}
f^{\circ}\left(x_{i} ; v_{i}\right) - \frac{1}{i}
&= \limsup _{y \rightarrow x_{i}, t \rightarrow 0} \frac{f(y+t v_{i})-f(y)}{t} - \frac{1}{i}\\
&\leq 
\frac{f\left(y_{i}+t_{i} v_{i}\right)-f\left(y_{i}\right)}{t_{i}}  \\
\end{align}
I would appreciate a lot if someone could explain to me what the thoughts behind this are.
 A: This is tricky! For me, the appearance of $1/i$ strongly suggested an approach that ultimately leads nowhere. Here's one way to work it out:
Define a family of functions, one for each $i \in \mathbb{N}$, by
$$
h_{i}\left(w,s\right) :=
\frac{f\left(w+sv_{i}\right) - f\left(w\right)}
{s}
$$
and consider the limit superior,
$$
f^{\circ}\left(x_{i},v_{i}\right)
:=
\limsup\limits_{w \to x_{i}, s \to 0}
\frac{f\left(w+sv_{i}\right) - f\left(w\right)}
{s}
=
\limsup\limits_{w \to x_{i}, s \to 0} h_{i}\left(w,s\right).
$$
Since the $\limsup$ is the largest of the limit points of
the sequences that have the form $h_{i}\left(w_{j},s_{j}\right)$ and
where $\left(w_{j},s_{j}\right)$ approaches $\left(x_{i},0\right)$,
there must be at least one such sequence $\left(w,s\right)_{j} =
\left(w_{j},s_{j}\right)$ making $h_{i}\left(w_{j},s_{j}\right)$
approach $f^{\circ}\left(x_{i},v_{i}\right)$. For that particular
sequence, we have
$$
\lim_{j \to \infty} h_{i}\left(w_{j},s_{j}\right)
=
f^{\circ}\left(x_{i},v_{i}\right).
$$
First note that we can go far enough down the sequence
$\left(w,s\right)_{j}$ so that $\left\lVert w_{j} - x_{i}\right\rVert
+ s_{j}$ is as close to zero as we'd like; this works because
$\left(w,s\right)_{j}$ was a sequence approaching
$\left(x_{i},0\right)$. We can also go far enough down the sequence to
make $h_{i}\left(w_{j},s_{j}\right)$ as close to its limit
$f^{\circ}\left(x_{i},v_{i}\right)$ as we'd like.
In particular, for each $k \in \mathbb{N}$, we
can let $j_{k}$ be the first index such that
\begin{gather*}
  \left\lVert w_{j_{k}} - x_{i}\right\rVert+ s_{j_{k}} < \frac{1}{k}\\
  \text{ and }\\
  f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{k} \le h_{i}\left(w_{j_{k}},s_{j_{k}}\right).
\end{gather*}
By taking $k=i$, this gives
\begin{gather*}
  \left\lVert w_{j_{i}} - x_{i}\right\rVert+ s_{j_{i}} < \frac{1}{i}\\
  \text{ and }\\
  f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{i} \le h_{i}\left(w_{j_{i}},s_{j_{i}}\right).
\end{gather*}
Now all we have to do is define new sequences by $y_{i} :=
w_{j_{i}}$ and $t_{i} := s_{j_{i}}$. Then
\begin{gather*}
  \left\lVert y_{i} - x_{i}\right\rVert + t_{i} < \frac{1}{i}\\
  \text{ and }\\
  f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{i} \le h_{i}\left(y_{i},t_{i}\right)
  =
  \frac{f\left(y_{i}+t_{i}v_{i}\right) - f\left(y_{i}\right)}
  {t_{i}}.
\end{gather*}
