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  "body": "\n## Proof\n\nWe add the two error terms, and we scale one of them.\n\n1. Let $V$, $W$, $U$, $f$, $g$, $x$ and $c$ be as in the statement. Put $A = Df(x)$ and $B = Dg(x)$.\n2. Since $A$ and $B$ are bounded, there are $C_A, C_B \\ge 0$ with $\\|A h\\| \\le C_A \\|h\\|$ and $\\|B h\\| \\le C_B \\|h\\|$ for each $h \\in V$. This is the meaning of [[bounded|0F3K4C7XYW]].\n3. The map $A + B$ is linear, and $\\|(A + B) h\\| \\le \\|A h\\| + \\|B h\\| \\le (C_A + C_B) \\|h\\|$ by the [[triangle inequality|0FJZMBX0ZZ]]. So $A + B$ is bounded.\n4. The map $c A$ is linear, and $\\|c A h\\| = |c| \\, \\|A h\\| \\le |c| C_A \\|h\\|$. So $c A$ is bounded.\n5. For $h \\in V$ with $x + h \\in U$, put $r(h) = f(x + h) - f(x) - A h$ and $s(h) = g(x + h) - g(x) - B h$.\n6. Then $(f + g)(x + h) - (f + g)(x) - (A + B) h = r(h) + s(h)$, and $(c f)(x + h) - (c f)(x) - (c A) h = c \\, r(h)$.\n7. Let $\\varepsilon \u003e 0$. By the [[little-o form of the derivative|0MN882VJ4P]] for $f$ and for $g$, there is $\\delta \u003e 0$ with $\\|r(h)\\| \\le \\varepsilon \\|h\\|$ and $\\|s(h)\\| \\le \\varepsilon \\|h\\|$. This holds for $\\|h\\| \u003c \\delta$ with $x + h \\in U$. We take $\\delta$ as the smaller of the two numbers given by $f$ and by $g$.\n8. For these $h$, the triangle inequality gives $\\|r(h) + s(h)\\| \\le 2 \\varepsilon \\|h\\|$. Given $\\varepsilon' \u003e 0$, we can apply this with $\\varepsilon = \\varepsilon' / 2$. So $r(h) + s(h) = o(\\|h\\|)$.\n9. For these $h$, also $\\|c \\, r(h)\\| = |c| \\, \\|r(h)\\| \\le |c| \\varepsilon \\|h\\|$. Given $\\varepsilon' \u003e 0$, we can apply this with $\\varepsilon = \\varepsilon' / (|c| + 1)$. So $c \\, r(h) = o(\\|h\\|)$.\n10. By steps 3, 6 and 8 and the little-o form, $f + g$ is [[Fréchet differentiable|08EVNYZF42]] at $x$ with $D(f + g)(x) = A + B$.\n11. By steps 4, 6 and 9 and the little-o form, $c f$ is Fréchet differentiable at $x$ with $D(c f)(x) = c A$.\n\n## Idea\n\nThe error of a sum is the sum of the errors, and the error of $c f$ is $c$ times the error of $f$. Both operations keep an error that is small compared with $\\|h\\|$. The candidate derivatives are again bounded linear maps, so the definition applies.\n",
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