> [!Lemma] Lemma (the exponential strictly increases) > The [[Real Exponential Function|real exponential function]] is strictly [[Monotone Real Function|increasing]]. **Proof**: Let $x<y$. By [[Lower Bound for Real Exponential Function]], for all $x\in \mathbb{R}, \; \exp(x)\geq1+x.$ Thus we have $e^{y}=e^{y-x}e^{x}\geq(1+y-x)e^{x}>e^{x}.$ **Proof**: Follows from [[Positive Derivative Implies Strictly Increasing Real Function]] since by [[Derivative of Real Exponential Function]], $\frac{d}{dx}(e^{x})=e^{x}\geq 0$ for all $x\in \mathbb{R}.$