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world:boule
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world:boule [2025/10/03 09:05] rdouc [Proof] |
world:boule [2025/10/03 09:06] (current) rdouc [Proof] |
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| To order these distances strictly, it is now enough to check that, for every $r>0$, the set $A \cap [0,r]$ is finite. But the cardinal of $A \cap [0,r]$ corresponds to the number of points in $D \cap \overline{B(c,r)}$, which is a compact set containing no accumulation points. Hence it is finite. | To order these distances strictly, it is now enough to check that, for every $r>0$, the set $A \cap [0,r]$ is finite. But the cardinal of $A \cap [0,r]$ corresponds to the number of points in $D \cap \overline{B(c,r)}$, which is a compact set containing no accumulation points. Hence it is finite. | ||
| - | We can thus write $A = \{\rho_n \; ; \; n \geq 1\}$ with $\rho_n$ strictly increasing. | + | We can thus write $A = \{\rho_n \; ; \; n \geq 1\}$ with strictly increasing $(\rho_n)$. |
| For every $n \geq 1$, by choosing a radius $\rho$ such that $\rho_n < \rho < \rho_{n+1}$, the ball $B(c,\rho)$ contains exactly $n$ points of $D$. | For every $n \geq 1$, by choosing a radius $\rho$ such that $\rho_n < \rho < \rho_{n+1}$, the ball $B(c,\rho)$ contains exactly $n$ points of $D$. | ||
world/boule.1759475125.txt.gz ยท Last modified: 2025/10/03 09:05 by rdouc
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