As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that **the derivative of $ f $ at $ a $ equals $ L $** if and only if
$$
\lim\_{x \to a} \frac{f(x) - f(a)}{x - a} = L.
$$

Now, most textbooks on particle mechanics offer the following recipe for defining the instantaneous velocity of a particle traveling along a straight line.

* Let $ f: \mathbb{R} \to \mathbb{R} $ denote the displacement function of a particle traveling along the $ x $-axis, with respect to time. The goal is to *define* the instantaneous velocity of the particle at time $ t\_{0} $.
* Pick a sequence of non-degenerate bounded closed intervals $ ([a\_{n},b\_{n}])\_{n \in \mathbb{N}} $ such that $ t\_{0} \in (a\_{n},b\_{n}) $ for all $ n \in \mathbb{N} $ and $ \displaystyle \lim\_{n \to \infty} (b\_{n} - a\_{n}) = 0 $.
* Compute the sequence of average velocities $ (v\_{\text{ave},n})\_{n \in \mathbb{N}} $ of the particle over these closed intervals, i.e.,
$$
(v\_{\text{ave},n})\_{n \in \mathbb{N}} \stackrel{\text{def}}{=} \left( \frac{f(a\_{n}) - f(b\_{n})}{a\_{n} - b\_{n}} \right)\_{n \in \mathbb{N}}.
$$
* Define the instantaneous velocity of the particle at time $ t\_{0} $ as $ \displaystyle \lim\_{n \to \infty} v\_{\text{ave},n} $, if it exists.

This recipe is somehow proposing a new definition of the derivative of a function. Let me describe it in precise mathematical terms.

> 
> Consider $ f: \mathbb{R} \to \mathbb{R} $. Let $ a \in \mathbb{R} $, and let $ \Lambda $ be the set of non-degenerate bounded closed intervals $ I $ such that $ a \in I^{\circ} $. Define a partial ordering $ \preceq $ on $ \Lambda $ by $ I \preceq J \iff I \supseteq J $ for all $ I,J \in \Lambda $. One can easily verify that $ (\Lambda,\preceq) $ is a directed set. Next, define a net $ \lambda: \Lambda \to \mathbb{R} $ by
>  $$
> \forall I \in \Lambda: \quad \lambda(I) \stackrel{\text{def}}{=} \frac{f({\frak{l}}(I)) - f({\frak{r}}(I))}{{\frak{l}}(I) - {\frak{r}}(I)},
> $$
>  where $ {\frak{l}}(I) $ and $ {\frak{r}}(I) $ denote the left- and right-endpoints of $ I $ respectively. Then define
>  $$
> f'(a) \stackrel{\text{def}}{=} \lim\_{I \in \Lambda} \lambda(I),
> $$
>  if it exists.
> 
> 
> 

**Question:** How can we show that this new definition of the derivative agrees with the usual one?