The paper presents a general bound on the dimensions of scalar operators in 4D conformal field theories (CFTs). For a scalar primary operator $\phi$ with dimension $d > 1$, the lowest dimension scalar that appears in the operator product expansion (OPE) of $\phi \times \phi$ is denoted $\Delta_{\min}$. The paper derives a model-independent inequality $\Delta_{\min} \leq f(d)$, where $f(d)$ is a continuous function that satisfies $f(1) = 2$. This bound is derived using general properties of CFTs, including unitarity, OPE, conformal block decomposition, and crossing symmetry. The function $f(d)$ is numerically evaluated and shown to approach 2 as $d \to 1$, indicating that the free theory limit is approached continuously. The bound is tested against known CFTs and is found to be satisfied by all weakly coupled 4D conformal fixed points. However, the Wilson-Fischer fixed points violate the bound by a constant factor, which is attributed to the subtleties of extrapolating to $4 - \varepsilon$ dimensions. The method is also applied to derive an analogous bound in 2D, where the Minimal Models satisfy the bound, with the Ising model nearly saturating it. The paper also discusses the phenomenological motivation for studying this problem, particularly in the context of constructing models of dynamical Electroweak Symmetry Breaking without flavor problems. The results are summarized in a sum rule derived from crossing symmetry, which is shown to be consistent with the bound. The paper concludes that the derived bound is a necessary condition for unitary CFTs and provides a framework for further study of CFTs in various dimensions.The paper presents a general bound on the dimensions of scalar operators in 4D conformal field theories (CFTs). For a scalar primary operator $\phi$ with dimension $d > 1$, the lowest dimension scalar that appears in the operator product expansion (OPE) of $\phi \times \phi$ is denoted $\Delta_{\min}$. The paper derives a model-independent inequality $\Delta_{\min} \leq f(d)$, where $f(d)$ is a continuous function that satisfies $f(1) = 2$. This bound is derived using general properties of CFTs, including unitarity, OPE, conformal block decomposition, and crossing symmetry. The function $f(d)$ is numerically evaluated and shown to approach 2 as $d \to 1$, indicating that the free theory limit is approached continuously. The bound is tested against known CFTs and is found to be satisfied by all weakly coupled 4D conformal fixed points. However, the Wilson-Fischer fixed points violate the bound by a constant factor, which is attributed to the subtleties of extrapolating to $4 - \varepsilon$ dimensions. The method is also applied to derive an analogous bound in 2D, where the Minimal Models satisfy the bound, with the Ising model nearly saturating it. The paper also discusses the phenomenological motivation for studying this problem, particularly in the context of constructing models of dynamical Electroweak Symmetry Breaking without flavor problems. The results are summarized in a sum rule derived from crossing symmetry, which is shown to be consistent with the bound. The paper concludes that the derived bound is a necessary condition for unitary CFTs and provides a framework for further study of CFTs in various dimensions.