Theorem: Generalized Exponential Limit

Let $x \in \R$ be a real number.
Let $\sequence {\nu_n}$ be an increasing, unbounded above real sequence.
Let $\sequence {\xi_n}$ be a real sequence such that:
:$\ds \lim_{n \mathop \to \infty} \nu_n \xi_n = x$
Then:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n} = \map \exp x$
where $\exp$ denotes the real exponential function defined as a power series.

Proof:
First, let us consider the case where all $\nu_n \in \Z_{\ge 0}$.
Then, by the Binomial Theorem for Integral Index:
:$\ds \paren 1 \quad \paren {1 + \xi_n}^{\nu_n} = \sum_{r \mathop = 0}^{\nu_n} \binom {\nu_n} r \xi_n^r$
Now, by Convergent Real Sequence is Bounded, there is some $X \in \R$ such that:
:$\size {\nu_j \xi_j} \le X$
for all $j \in \N$.
We will apply Tannery's Theorem, using:
:$p_n = \nu_n$
:$\map {v_r} n = \dbinom {\nu_n} r \xi_n^r$
:$w_r = \dfrac {x^r} {r!}$
:$M_r = \dfrac {X^r} {r!}$
For $r \le p_n = \nu_n$, we have that:
:$\paren 2 \quad \map {v_r} n = \dfrac {\dbinom {\nu_n} r} {\nu_n^r} \paren {\nu_n \xi_n}^r$
and by Binomial Coefficient over Power Not Greater than Reciprocal of Factorial:
:$\dfrac {\dbinom {\nu_n} r} {\nu_n^r} \le \dfrac 1 {r!}$
from which it immediately follows that:
:$\size {\map {v_r} n} = \dfrac {\dbinom {\nu_n} r} {\nu_n^r} \size {\nu_n \xi_n}^r \le M_r$
By Radius of Convergence of Power Series over Factorial, we see that:
:$\sum_{r \mathop = 0}^\infty M_r$ is convergent
Finally:
{{begin-eqn}}
{{eqn | l = \lim_{n \to \infty} \map {v_r} n
      | r = \lim_{n \to \infty} \frac {\binom {\nu_n} r} {\nu_n^r} \paren {\nu_n \xi_n}^r
      | c = By $\paren 2$, for sufficiently large $n$
}}
{{eqn | r = \lim_{n \to \infty} \frac {\binom {\nu_n} r} {\nu_n^r} \cdot \paren {\lim_{n \to \infty} \nu_n \xi_n}^r
      | c = Product Rule for Real Sequences
}}
{{eqn | r = w_r
      | c = Limit to Infinity of Binomial Coefficient over Power and hypothesis on $\sequence {\xi_n}$
}}
{{end-eqn}}
Since the conditions are satisfied, we conclude that:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n}
      | r = \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{\nu_n} \binom {\nu_n} r \xi_n^r
      | c = By $\paren 1$
}}
{{eqn | r = \lim_{n \mathop \to \infty} \sum_{r \mathop = 0}^{p_n} \map {v_r} n
      | c = Definitions of $p_n$ and $v_r$
}}
{{eqn | r = \sum_{r \mathop = 0}^\infty w_r
      | c = Tannery's Theorem
}}
{{eqn | r = \exp x
      | c = Definition of Real Exponential Function as a power series
}}
{{end-eqn}}{{qed|lemma}}
Now, we move to the general case for $\nu_n$.
{{WLOG}}, assume that $\nu_n \ge 0$.
First, we must observe that $\xi_n \to 0$ as $n \to \infty$.
This follows by the Squeeze Theorem from:
:$0 \le \size {\xi_n} \le \nu_n \size {\xi_n}$
which holds for $n$ such that $\nu_n \ge 1$.
Define two sequences $\sequence {\underline \nu_n}, \sequence {\overline \nu_n}$ as:
:$\underline \nu_n = \floor {\nu_n}$
:$\overline \nu_n = \floor {\nu_n} + 1$
Then, we have:
:$\nu_n - 1 \le \underline \nu_n \le \overline \nu_n \le \nu_n + 1$
and so:
:$\nu_n \xi_n - \size {\xi_n} \le \underline \nu_n \xi_n, \overline \nu_n \xi_n \le \nu_n \xi_n + \size {\xi_n}$
hence by the Squeeze Theorem:
:$\underline \nu_n \xi_n \to x$
:$\overline \nu_n \xi_n \to x$
By the special case already determined, we thus have:
:$\paren {1 + \xi_n}^{\underline \nu_n} \to \map \exp x$
:$\paren {1 + \xi_n}^{\overline \nu_n} \to \map \exp x$
But, by Power Function is Monotone over Reals:
:$\map \min {\paren {1 + \xi_n}^{\underline \nu_n}, \paren {1 + \xi_n}^{\overline \nu_n}} \le \paren {1 + \xi_n}^{\nu_n} \le \map \max {\paren {1 + \xi_n}^{\underline \nu_n}, \paren {1 + \xi_n}^{\overline \nu_n}}$
Hence, once more by the Squeeze Theorem:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \xi_n}^{\nu_n} = \map \exp x$
{{qed}}