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For real square-integrable $f,g$, nonnegativity of $\int (f-\lambda g{)}^{2}dx$ gives a quadratic in the real number $\lambda$. If $\int g^{2}dx>0$, choose $\lambda =\int fgdx/\int g^{2}dx$ to obtainIf $\int g^{2}dx=0$, then $g=0$ almost everywhere and the same inequality is immediate. This proves the integral Cauchy-Schwarz inequality.

Put $T=t_f-t_i>0$ and $L=x_f-x_i>0$. The forward speed satisfies $L={\int }_{t_i}^{t_f}vdt$, so $V_t=L/T$. The change of variables $dx=vdt$ gives $V_x=L^{-1}{\int }_{t_i}^{t_f}{v}^{2}dt$. Apply the Cauchy-Schwarz inequality to $v$ and the constant function one:Equality holds exactly when $v$ is constant almost everywhere. Equivalently, the time and distance averages of speed satisfy $V_x-V_t={\mathrm{Var}}_t(v)/V_t$, which identifies the excess as a measure of speed variation.