Sequential Criterion for Differentiability at an Interior Point

Let $I$ be an interval, let $f:I\to\mathbb{R}$, let $x_0\in I$ be an interior point of $I$, and let $L$ be a real number. Call a sequence $(h_k)_{k\in\mathbb{N}}$ of real numbers admissible if $h_k\ne0$ and $x_0+h_k\in I$ for every $k$, and $(h_k)_{k\in\mathbb{N}}$ has limit $0$. For an admissible sequence write

$$q_k=\frac{f(x_0+h_k)-f(x_0)}{h_k}.$$

1. If $f$ is differentiable at $x_0$ with $f'(x_0)=L$, then $(q_k)_{k\in\mathbb{N}}$ converges to $L$ for every admissible sequence.

2. Conversely, if $(q_k)_{k\in\mathbb{N}}$ converges to $L$ for every admissible sequence, then $f$ is differentiable at $x_0$ and $f'(x_0)=L$.