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  "body": "\n## Proof\n\nWith the max norm, the error of $F$ is small exactly when both errors are small.\n\n1. Let $V$, $W_1$, $W_2$, $U$, $x$, $f_1$, $f_2$ and $F$ be as in the statement. Put $A_1 = Df_1(x)$ and $A_2 = Df_2(x)$.\n2. Define $A \\colon V \\to W_1 \\times W_2$ by $A h = (A_1 h, A_2 h)$. It is linear, since $A_1$ and $A_2$ are linear and the vector operations on $W_1 \\times W_2$ act on each factor.\n3. Since $A_1$ and $A_2$ are [[bounded|0F3K4C7XYW]], there are $C_1, C_2 \\ge 0$ with $\\|A_i h\\| \\le C_i \\|h\\|$ for $i = 1, 2$ and each $h \\in V$.\n4. So $\\|A h\\| = \\max(\\|A_1 h\\|, \\|A_2 h\\|) \\le \\max(C_1, C_2) \\|h\\|$, and $A$ is bounded.\n5. For $h \\in V$ with $x + h \\in U$, put $r_i(h) = f_i(x + h) - f_i(x) - A_i h$ for $i = 1, 2$.\n6. Then $F(x + h) - F(x) - A h = (r_1(h), r_2(h))$. Its norm is $\\max(\\|r_1(h)\\|, \\|r_2(h)\\|)$.\n7. Let $\\varepsilon \u003e 0$. By the [[little-o form of the derivative|0MN882VJ4P]] for $f_1$ and $f_2$, there are $\\delta_1, \\delta_2 \u003e 0$ with $\\|r_i(h)\\| \\le \\varepsilon \\|h\\|$ for $\\|h\\| \u003c \\delta_i$ and $x + h \\in U$.\n8. Put $\\delta = \\min(\\delta_1, \\delta_2)$. For $\\|h\\| \u003c \\delta$ with $x + h \\in U$, step 6 gives $\\|F(x + h) - F(x) - A h\\| \\le \\varepsilon \\|h\\|$.\n9. So $F(x + h) - F(x) - A h = o(\\|h\\|)$. By the little-o form, $F$ is [[Fréchet differentiable|08EVNYZF42]] at $x$ with $DF(x) = A$.\n10. That is, $DF(x)(h) = (Df_1(x)(h), Df_2(x)(h))$ for each $h \\in V$.\n\n## Idea\n\nThe error of the pair map is the pair of the two errors. In the max norm, a pair is small exactly when both of its entries are small. So the pair of the two derivatives is a derivative of the pair map.\n",
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