Derivative of Exponential Function/Proof 5

Proof
This proof assumes the limit definition of $\exp$.

So let:
 * $\displaystyle \forall n \in \N: \forall x \in \R: f_n \left({x }\right) = \left({1 + \frac x n}\right)^n$

Let $x_0 \in \R$.

Consider $I := \left[{x_0 - 1, \,.\,.\, x_0 + 1}\right]$.

Let:
 * $N = \left\lceil{\max \left\{ {\left\vert{x_0 - 1}\right\vert, \left\vert{x_0 + 1}\right\vert} \right\} }\right\rceil$

where $\left\lceil{ \cdot }\right\rceil$ denotes the ceiling function.

From Closed Real Interval is Compact, $I$ is compact.

From Chain Rule:
 * $\displaystyle D_x f_n \left({x}\right) = \frac n {n + x} f_n \left({x}\right)$

Lemma
From the lemma:
 * $\displaystyle \forall x \in I : \left\langle{D_x f_{n + N}\left({x}\right)}\right\rangle$ is increasing

Hence, from Dini's Theorem, $\left\langle{ D_x f_{n + N} }\right\rangle$ is uniformly convergent on $I$.

Therefore, for $x \in I$:

In particular:
 * $D_x \exp x_0 = \exp x_0$