= Factorial estimate for Picard iteration
{title2=$|y_{n+1}(x)-y_n(x)|\leq ML^n|x-a|^{n+1}/(n+1)!$}
For <Picard iteration> $y_{n+1}(x)=b+\int_a^x f(s,y_n(s))\,ds$, $y_0=b$, assume $|f|\leq M$ and a <Lipschitz constant> $L$ on a rectangle containing every iterate. The first difference is at most $M|x-a|$. Integrating the preceding bound multiplied by $L$ proves the displayed estimate by <mathematical induction>. Its uniform bounds sum to a convergent <exponential series> for every finite $Lh$, so the <Weierstrass M-test> gives <uniform convergence> without requiring $Lh<1$.
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