Proof of Gronwall's Lemma (Integral Form)

Assume $u(t)\le\alpha+\beta\int_{a}^{t}u$ for every $t\in[a,b]$. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; derivatives of maps $[a,b]\to\mathbb{R}$ are taken relative to $[a,b]$, as in Derivative of a Real Function on an Interval (i).

The integral. By A Continuous Function on a Closed Interval is Riemann Integrable (i), $u$ is bounded and integrable on $[a,b]$. Let $U:[a,b]\to\mathbb{R}$ be the map with $U(t)=\int_{a}^{t}u$ for $t\in[a,b]$, of Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (i). Since $u$ is continuous at every $t\in[a,b]$, Fundamental Theorem of Calculus, Part I: the Indefinite Integral is Lipschitz, Continuous, and Differentiable at Points of Continuity (iv) shows that $U$ is differentiable at $t$ with $U'(t)=u(t)$, for every $t\in[a,b]$. Moreover $U(a)=\int_{a}^{a}u=0$ by The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii).

The weight. Let $h:[a,b]\to\mathbb{R}$ be the map $t\mapsto-\beta(t-a)=(-\beta)t+\beta a$ (the map $(-\beta)\,\mathrm{id}_{[a,b]}+k$, with the identity map of Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii), read as a map into $\mathbb{R}$, the constant map $k$ with value $\beta a$ of Pointwise Operations on Real-Valued Maps (ii), and the scalar multiple and sum of Pointwise Operations on Real-Valued Maps (i)). By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv), $h$ is differentiable at every $t\in[a,b]$ with $h'(t)=-\beta$. By Basic Facts about Intervals of the Real Line (i), $\mathbb{R}$ is an interval containing two distinct points, and by Basic Properties of the Exponential Function (vii) the map $\exp$ (The Real Exponential Function (ii), with sums of series as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii)) is differentiable at every $s\in\mathbb{R}$ with derivative $\exp(s)$ (that clause, with $\mathbb{R}$ in place of $I$, is about $\exp|_{\mathbb{R}}$, which is $\exp$ by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ix)). By Chain Rule for Real Functions on Intervals (i), with $[a,b]$ and $\mathbb{R}$ in place of $I$ and $J$, $h$ in place of $f$ and $\exp$ in place of $g$, the map $w=\exp\circ h:[a,b]\to\mathbb{R}$, $w(t)=\exp\bigl(-\beta(t-a)\bigr)$, is differentiable at every $t\in[a,b]$ with

$$w'(t)=\exp\bigl(h(t)\bigr)\,(-\beta)=-\beta\,w(t).$$

For $t\in[a,b]$, $w(t)>0$ and $\exp\bigl(\beta(t-a)\bigr)>0$ by Basic Properties of the Exponential Function (iii), and

$$\exp\bigl(\beta(t-a)\bigr)\,w(t)=\exp\bigl(\beta(t-a)-\beta(t-a)\bigr)=\exp(0)=1$$

by Basic Properties of the Exponential Function (ii) and Basic Properties of the Exponential Function (i); in particular $w(a)=\exp(0)=1$.

A nonincreasing function. Let $R:[a,b]\to\mathbb{R}$ be the map $t\mapsto\alpha+\beta U(t)$, the sum of the constant map with value $\alpha$ and the scalar multiple $\beta U$. By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv), $R$ is differentiable at every $t\in[a,b]$ with $R'(t)=\beta u(t)$. Let $\psi=wR:[a,b]\to\mathbb{R}$ be the pointwise product. By Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), $\psi$ is differentiable at every $t\in[a,b]$ with

$$\psi'(t)=w'(t)R(t)+w(t)R'(t)=-\beta w(t)R(t)+\beta w(t)u(t)=\beta\,w(t)\bigl(u(t)-R(t)\bigr).$$

By the hypothesis, $u(t)-R(t)\le0$; since $\beta\ge0$ and $w(t)>0$, we get $\psi'(t)\le0$ for every $t\in[a,b]$. By The Increment Form of the Derivative, and Differentiability Implies Continuity (ii), $\psi$ is continuous at every $t\in[a,b]$ relative to $[a,b]$, so it is continuous on $[a,b]$ by Continuous Map Between Metric Spaces (ii). Hence The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (iii), with $[a,b]$ in place of $I$ and $\psi$ in place of $f$, gives, for every $t\in[a,b]$ (so $a\le t$),

$$\psi(t)\le\psi(a)=w(a)\bigl(\alpha+\beta U(a)\bigr)=1\cdot(\alpha+\beta\cdot0)=\alpha .$$

Conclusion. Let $t\in[a,b]$. Multiplying $w(t)R(t)=\psi(t)\le\alpha$ by $\exp\bigl(\beta(t-a)\bigr)>0$ and using $\exp\bigl(\beta(t-a)\bigr)w(t)=1$ gives $R(t)\le\alpha\exp\bigl(\beta(t-a)\bigr)$. By the hypothesis, $u(t)\le R(t)$, and therefore

$$u(t)\le\alpha\exp\bigl(\beta(t-a)\bigr),$$

which is (i).