Proof of Uniqueness of the Partial Derivative on an Open Subset of $\mathbb{R}^n$

Throughout, the arithmetic and order of $\mathbb{R}$ are used through The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (v) and Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix), and norms in $\mathbb{R}^{n}$ through Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v) and Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length (iii). For a real $t\neq0$ with $a+te_{i}\in U$ write

$$q(t)=\frac{f(a+te_{i})-f(a)}{t}$$

for the difference quotient.

Suppose, for a contradiction, that $L\neq M$, and put $\varepsilon=|L-M|/2$, a positive real number. The hypothesis for $N=L$ gives a real $\delta_{L}>0$, and the hypothesis for $N=M$ a real $\delta_{M}>0$, each for this $\varepsilon$. Let $\delta=\min\{\delta_{L},\delta_{M}\}>0$ and, having chosen $\delta$, take $t=\delta/2$. Then $0<|t|<\delta_{L}$ and $0<|t|<\delta_{M}$, so both properties apply to $t$: the point $a+te_{i}$ lies in $U$, the quotient $q(t)$ is defined, and

$$|q(t)-L|<\varepsilon,\qquad |q(t)-M|<\varepsilon .$$

By the triangle inequality for the absolute value, Rules of Arithmetic and Order in an Ordered Field (xi),

$$|L-M|\le|L-q(t)|+|q(t)-M|<2\varepsilon=|L-M| ,$$

which is impossible. Hence $L=M$, which is (i).