**Strategy:** We first prove the inequality for the case $p=1$ using the Fundamental Theorem of Calculus and the Generalized Hölder Inequality. We then derive the general case $1 < p < n$ by applying the $p=1$ result to a power of $|u|$. Finally, we rely on the density of $C_c^\infty(\mathbb{R}^n)$ in $W^{1,p}(\mathbb{R}^n)$ to extend the result to all Sobolev functions.

**Step 1: The Case $p=1$** Assume $u \in C_c^1(\mathbb{R}^n)$. Since $u$ has compact support, $u(x) \to 0$ as $|x| \to \infty$. For any fixed $x \in \mathbb{R}^n$ and any index $i \in \{1, \dots, n\}$, we express $u(x)$ as the integral of its partial derivative $\partial_i u$ along the $i$-th coordinate axis: \begin{align*} u(x) = \int_{-\infty}^{x_i} \partial_i u(x_1, \dots, t, \dots, x_n) \, \mathrm{d}t. \end{align*} Taking the absolute value and bounding the integral by the integral over the entire real line: \begin{align*} |u(x)| \le \int_{-\infty}^{\infty} |\partial_i u(x_1, \dots, y_i, \dots, x_n)| \, \mathrm{d}y_i. \end{align*} This inequality holds for each $i = 1, \dots, n$. We raise both sides to the power $\frac{1}{n-1}$ and take the product over all $i=1, \dots, n$: \begin{align*} |u(x)|^{\frac{n}{n-1}} \le \prod_{i=1}^n \left( \int_{-\infty}^{\infty} |\partial_i u(x)| \, \mathrm{d}y_i \right)^{\frac{1}{n-1}}. \end{align*} Note: In the integral for index $i$, the variable $x_i$ is replaced by the integration variable $y_i$. We define the Sobolev conjugate for $p=1$ as $1^* = \frac{n}{n-1}$. We now integrate this inequality with respect to $x_1$ over $\mathbb{R}$. \begin{align*} \int_{-\infty}^\infty |u(x)|^{\frac{n}{n-1}} \, \mathrm{d}x_1 \le \int_{-\infty}^\infty \left[ \left( \int_{-\infty}^{\infty} |\partial_1 u| \, \mathrm{d}y_1 \right)^{\frac{1}{n-1}} \prod_{i=2}^n \left( \int_{-\infty}^{\infty} |\partial_i u| \, \mathrm{d}y_i \right)^{\frac{1}{n-1}} \right] \, \mathrm{d}x_1. \end{align*} Observe that the first factor $I_1 = \left( \int_{-\infty}^{\infty} |\partial_1 u| \, \mathrm{d}y_1 \right)^{\frac{1}{n-1}}$ does **not** depend on $x_1$. We can pull it outside the integral: \begin{align*} \int_{-\infty}^\infty |u(x)|^{\frac{n}{n-1}} \, \mathrm{d}x_1 \le I_1 \int_{-\infty}^\infty \prod_{i=2}^n \left( \int_{-\infty}^{\infty} |\partial_i u| \, \mathrm{d}y_i \right)^{\frac{1}{n-1}} \, \mathrm{d}x_1. \end{align*} We apply the **Generalized Hölder Inequality** to the integral of the product. There are $n-1$ terms in the product (indices $i=2, \dots, n$). We choose the exponent $r = n-1$ for each function, so that $\sum_{i=2}^n \frac{1}{r} = \frac{n-1}{n-1} = 1$. \begin{align*} \int_{-\infty}^\infty \prod_{i=2}^n (\dots)^{\frac{1}{n-1}} \, \mathrm{d}x_1 \le \prod_{i=2}^n \left( \int_{-\infty}^\infty \int_{-\infty}^\infty |\partial_i u| \, \mathrm{d}y_i \, \mathrm{d}x_1 \right)^{\frac{1}{n-1}}. \end{align*} We now proceed by induction, integrating successively with respect to $x_2, x_3, \dots, x_n$. At each step $k$, the factor involving $\partial_k u$ comes out of the integral (as it depends on $y_k$ and not $x_k$), and we apply Hölder's inequality to the remaining terms. After integrating over all variables $x_1, \dots, x_n$, we obtain: \begin{align*} \int_{\mathbb{R}^n} |u(x)|^{\frac{n}{n-1}} \, \mathrm{d}\mathcal{L}^n(x) \le \prod_{i=1}^n \left( \int_{\mathbb{R}^n} |\partial_i u(x)| \, \mathrm{d}\mathcal{L}^n(x) \right)^{\frac{1}{n-1}}. \end{align*} Taking the $(n-1)/n$ power of both sides: \begin{align*} \|u\|_{L^{\frac{n}{n-1}}} \le \left( \prod_{i=1}^n \|\partial_i u\|_{L^1} \right)^{\frac{1}{n}}. \end{align*} Using the Arithmetic Mean-Geometric Mean inequality ($\sqrt[n]{\prod a_i} \le \frac{1}{n} \sum a_i$) and the equivalence of norms on $\mathbb{R}^n$ ($\sum |z_i| \le \sqrt{n} |z|$): \begin{align*} \|u\|_{L^{\frac{n}{n-1}}} \le \frac{1}{n} \sum_{i=1}^n \|\partial_i u\|_{L^1} \le \frac{1}{n} \sqrt{n} \|\nabla u\|_{L^1} = C \|\nabla u\|_{L^1}. \end{align*} This proves the theorem for $p=1$.

**Step 2: The General Case $1 < p < n$** Let $u \in C_c^1(\mathbb{R}^n)$. We apply the $p=1$ inequality to the function $v := |u|^\gamma$, where $\gamma > 1$ is a constant to be determined. First, we compute the gradient of $v$. By the Chain Rule: \begin{align*} \nabla v = \gamma |u|^{\gamma-1} (\operatorname{sgn} u) \nabla u \implies |\nabla v| = \gamma |u|^{\gamma-1} |\nabla u|. \end{align*} Applying the $p=1$ GNS inequality to $v$: \begin{align*} \|v\|_{L^{\frac{n}{n-1}}} \le C \|\nabla v\|_{L^1}. \end{align*} Substituting the definitions of $v$ and $\nabla v$: \begin{align*} \left( \int_{\mathbb{R}^n} |u|^{\frac{\gamma n}{n-1}} \, \mathrm{d}\mathcal{L}^n \right)^{\frac{n-1}{n}} \le C \gamma \int_{\mathbb{R}^n} |u|^{\gamma-1} |\nabla u| \, \mathrm{d}\mathcal{L}^n. \end{align*} We choose $\gamma$ such that the exponent on the left matches $p^*$. We set: \begin{align*} \frac{\gamma n}{n-1} = p^* = \frac{np}{n-p} \implies \gamma = \frac{p(n-1)}{n-p}. \end{align*} Since $p < n$, we verify that $\gamma > 1$: \begin{align*} \gamma = \frac{pn - p}{n - p} > \frac{pn - p \cdot p}{n - p} = p > 1. \end{align*} Now we estimate the RHS integral using Hölder's Inequality with exponents $p$ and its conjugate $p' = p/(p-1)$: \begin{align*} \int_{\mathbb{R}^n} |u|^{\gamma-1} |\nabla u| \, \mathrm{d}\mathcal{L}^n \le \left( \int_{\mathbb{R}^n} |u|^{(\gamma-1) \frac{p}{p-1}} \, \mathrm{d}\mathcal{L}^n \right)^{\frac{p-1}{p}} \left( \int_{\mathbb{R}^n} |\nabla u|^p \, \mathrm{d}\mathcal{L}^n \right)^{\frac{1}{p}}. \end{align*} We verify the exponent of $|u|$ in the first integral: \begin{align*} (\gamma - 1) \frac{p}{p-1} &= \left( \frac{p(n-1)}{n-p} - 1 \right) \frac{p}{p-1} \\ &= \frac{pn - p - (n-p)}{n-p} \cdot \frac{p}{p-1} \\ &= \frac{pn - n}{n-p} \cdot \frac{p}{p-1} \\ &= \frac{n(p-1)}{n-p} \cdot \frac{p}{p-1} = \frac{np}{n-p} = p^*. \end{align*} Thus, the inequality becomes: \begin{align*} \|u\|_{L^{p^*}}^{\gamma} \le C \gamma \|u\|_{L^{p^*}}^{\gamma-1} \|\nabla u\|_{L^p}. \end{align*} Since $u$ has compact support, $\|u\|_{L^{p^*}}$ is finite. We can divide both sides by $\|u\|_{L^{p^*}}^{\gamma-1}$: \begin{align*} \|u\|_{L^{p^*}} \le C \gamma \|\nabla u\|_{L^p}. \end{align*}

**Step 3: Density Argument** Since $C_c^\infty(\mathbb{R}^n)$ is dense in $W^{1,p}(\mathbb{R}^n)$, for any $u \in W^{1,p}(\mathbb{R}^n)$, there exists a sequence $\{u_m\} \subset C_c^\infty(\mathbb{R}^n)$ converging to $u$ in the $W^{1,p}$ norm. The inequality holds for each $u_m$: \begin{align*} \|u_m\|_{L^{p^*}} \le C \|\nabla u_m\|_{L^p}. \end{align*} Taking the limit as $m \to \infty$, both norms converge to the norms of $u$ (continuity of norms). Thus the inequality holds for $u$.