Proof of Sequential Criterion for Differentiability at an Interior Point

Throughout, differentiability at $x_0$ with derivative $L$ means, as in Derivative at an Interior Point, that for every real $\varepsilon>0$ there is a real $\delta>0$ such that every real $h$ with $0<|h|<\delta$ and $x_0+h\in I$ satisfies

$$\left|\frac{f(x_0+h)-f(x_0)}{h}-L\right|<\varepsilon .$$

Claim 1. Let $(h_k)_{k\in\mathbb{N}}$ be admissible and let $\varepsilon>0$. Differentiability supplies $\delta>0$ as above. Since $(h_k)_{k\in\mathbb{N}}$ has limit $0$, there is $K\in\mathbb{N}$ with $|h_k|<\delta$ for every $k\ge K$; for such $k$ we also have $h_k\ne0$ and $x_0+h_k\in I$, so $0<|h_k|<\delta$ and therefore $|q_k-L|<\varepsilon$. As $\varepsilon$ was arbitrary, $(q_k)_{k\in\mathbb{N}}$ has limit $L$.

Claim 2. Suppose, for contradiction, that the displayed condition fails for the number $L$. Negating it, there is a real $\varepsilon>0$ such that for every real $\delta>0$ there is a real $h$ with $0<|h|<\delta$, with $x_0+h\in I$, and with

$$\left|\frac{f(x_0+h)-f(x_0)}{h}-L\right|\ge\varepsilon .$$

Let $(\delta_k)_{k\in\mathbb{N}}$ be a sequence of positive real numbers with limit $0$, which exists by Existence of a Sequence of Positive Real Numbers with Limit Zero. For each $k$ let $S_k$ be the set of real numbers $h$ satisfying $0<|h|<\delta_k$, $x_0+h\in I$, and the displayed inequality; by the previous paragraph, applied with $\delta=\delta_k$, each $S_k$ is nonempty. By Axiom of Countable Choice there is a sequence $(h_k)_{k\in\mathbb{N}}$ with $h_k\in S_k$ for every $k$.

This sequence is admissible. Indeed $h_k\ne0$ and $x_0+h_k\in I$ for every $k$; and given a real $\eta>0$, the limit of $(\delta_k)_{k\in\mathbb{N}}$ supplies $K$ with $\delta_k<\eta$ for every $k\ge K$, whence $|h_k-0|=|h_k|<\delta_k<\eta$ for those $k$, so $(h_k)_{k\in\mathbb{N}}$ has limit $0$.

On the other hand $|q_k-L|\ge\varepsilon$ for every $k$, so $(q_k)_{k\in\mathbb{N}}$ does not have limit $L$: taking $\varepsilon$ itself as the tolerance in Limit of a Sequence of Real Numbers, no index beyond which $|q_k-L|<\varepsilon$ exists. This contradicts the hypothesis, which applies to the admissible sequence just constructed.

Therefore the displayed condition holds for $L$, so $f$ is differentiable at $x_0$, and $L$ is its derivative there, the derivative being unique by Uniqueness of the Derivative at an Interior Point.