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  "body": "\n## Proof\n\nWe compare the norm with the sup norm of the coordinates and use completeness of $\\mathbb{R}$ in each coordinate.\n\n1. Let $\\mathbb{K}$ be $\\mathbb{R}$ or $\\mathbb{C}$. If $V = \\{0\\}$, each sequence is constant and converges, so $V$ is complete. Let $V \\neq \\{0\\}$.\n2. Since $V$ is [[finite-dimensional|018W3V3TCT]], it has a [[basis|06MRNP4BNT]] $e_1, \\ldots, e_n$. Write $x = \\sum_j a_j(x) e_j$, with linear coordinates $a_j$, by [[this result|059THXBRBB]].\n3. Let $\\|x\\|_\\infty = \\max_j |a_j(x)|$. It is a norm: it vanishes only when all coordinates vanish, it is homogeneous, and $|a_j(x + y)| \\le |a_j(x)| + |a_j(y)|$. By the [[equivalence of norms in finite dimension|03MC70ZRNR]], there are $c, C \u003e 0$ with $c \\|x\\|_\\infty \\le \\|x\\| \\le C \\|x\\|_\\infty$ for all $x$, by the definition of [[equivalent norms|0J3578562S]].\n4. Let $(x_k)$ be a [[Cauchy sequence|0MCK8K2D7Z]] in $(V, \\|\\cdot\\|)$. Fix $j$. Then $|a_j(x_k) - a_j(x_l)| \\le \\|x_k - x_l\\|_\\infty \\le \\|x_k - x_l\\| / c$. So $(a_j(x_k))_k$ is a Cauchy sequence in $\\mathbb{K}$.\n5. Its real parts and imaginary parts form Cauchy sequences in $\\mathbb{R}$, since $|\\operatorname{Re} z| \\le |z|$ and $|\\operatorname{Im} z| \\le |z|$.\n6. By the [[completeness of the real numbers|E3CPZNJZNG]], both converge. So $a_j(x_k) \\to \\alpha_j$ for some $\\alpha_j \\in \\mathbb{K}$.\n7. Let $x = \\sum_j \\alpha_j e_j$. Then $a_j(x_k - x) = a_j(x_k) - \\alpha_j$, so $\\|x_k - x\\| \\le C \\max_j |a_j(x_k) - \\alpha_j|$.\n8. Given $\\varepsilon \u003e 0$, for each $j$ there is $N_j$ with $|a_j(x_k) - \\alpha_j| \u003c \\varepsilon / C$ for $k \\ge N_j$. For $k \\ge \\max_j N_j$, we get $\\|x_k - x\\| \u003c \\varepsilon$.\n9. So $(x_k)$ [[converges|06JYPJ4WHH]] to $x$. Each Cauchy sequence converges, so $V$ is [[complete|08EWT7J0SC]], and $V$ is a [[Banach space|0BDTTM24DP]].\n\n## Idea\n\nAll norms on a finite-dimensional space are equivalent, so we may measure with the largest coordinate. Then a Cauchy sequence of vectors gives finitely many Cauchy sequences of numbers, which converge because $\\mathbb{R}$ is complete.\n",
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