We prove the three cases for $k = 1$ and deduce the general $k$ by induction. **Case 1: $k = 1$, $p < d$ (subcritical).** Let $u \in C_c^\infty(\mathbb{R}^d)$ and set $v := |u|^{t-1}u$ for $t \ge 1$ to be chosen. By the [fundamental theorem of calculus](/theorems/632), $|v(x)| \le \int_{-\infty}^{x_i} |\partial_{x_i} v| \, dy$ for each $i$, giving $|v(x)|^{d/(d-1)} \le \prod_{i=1}^d f_i(\tilde{x}_i)^{1/(d-1)}$ where $f_i(\tilde{x}_i) = \int_{\mathbb{R}} |\partial_{x_i} v| \, dx_i$. The Gagliardo-Nirenberg-Sobolev product inequality (integrating and applying generalised Hölder) yields $\|v\|_{d/(d-1)} \le \prod_{i=1}^d \|\partial_{x_i} v\|_1^{1/d}$. Since $\partial_{x_i} v = t|u|^{t-1} \partial_{x_i} u$, applying Hölder's inequality with exponents $p$ and $q = p/(p-1)$ and choosing $t = q(d-1)/d = (pd-p)/(d-p)$ to match exponents gives $\|u\|_{p^*} \le C \|u\|_{W^{1,p}}$ where $p^* = pd/(d-p)$. The general $q \in [p, p^*]$ follows by interpolation. Extension from $C_c^\infty$ to $W^{1,p}$ is by density. **Case 2: $k = 1$, $p = d$ (critical).** The same argument as Case 1, combined with the interpolation inequality $\|u\|_{d(t-1)/(d-1)} \le \|u\|_d^{\theta(t-1)} \cdot \|\nabla u\|_d^{t - (d-1)}$, yields $\|u\|_q \le C_q \|u\|_{W^{1,d}}$ for any $q \in [d, \infty)$. **Case 3: $k = 1$, $p > d$ (supercritical).** This is [Morrey's Inequality](/theorems/62). For $u \in C_c^\infty(\mathbb{R}^d)$ and a cube $Q = [-r, r]^d$ with average $\bar{u}$, the fundamental theorem of calculus and Hölder's inequality give $|u(y) - \bar{u}| \le Cr^{1-d/p} \|\nabla u\|_{L^p}$. The triangle inequality $|u(y_1) - u(y_2)| \le |u(y_1) - \bar{u}| + |u(y_2) - \bar{u}|$ with $r \sim |y_1 - y_2|$ gives the Hölder estimate. The $L^\infty$ bound follows from $|u(x)| \le |\bar{u}| + Cr^{1-d/p}\|\nabla u\|_{L^p}$ with $\bar{u}$ controlled by Hölder's inequality. Extension to $W^{1,p}$ by density. **General $k$.** Iterate: if $k < d/p$, apply Case 1 repeatedly (each application uses one derivative and raises the Lebesgue exponent from $p$ to $p^* = pd/(d-p)$). If after some iterations the exponent reaches or exceeds $d$, apply Case 2 or 3 to the remaining [derivatives](/page/Derivative).