\(d(x,A)=0\)であることは，定義によって，いかなる正数\(\epsilon\)を与えた場合にも，\(d(x,y)<\epsilon\)となる\(y\in A\)が存在すること（中略）を意味する．
えっなんで？
証明
\(\{d(x,y)|y \in A\}=M\)とおく．
\begin{align*}
&~d(x,A)=0\\
\Longleftrightarrow &~\inf\{d(x,y)|y \in A\}=0\\
\Longleftrightarrow &~\inf M=0\\
\Longleftrightarrow &~\begin{cases}\forall c \in M [0\leq c] \quad\text{※ 恒真命題}\\ \forall c \in M [c^{\prime} \leq c]\Rightarrow c^{\prime}\leq 0\end{cases}\\
\Longleftrightarrow &~\forall c \in M [c^{\prime} \leq c]\Rightarrow c^{\prime}\leq 0\\
\Longleftrightarrow &~\forall c^{\prime}\in \mathbb{R}\left[\forall c \in M [c^{\prime} \leq c]\rightarrow c^{\prime}\leq 0\right]\\
\Longleftrightarrow &~\forall c^{\prime}\in \mathbb{R}\left[\overline{\forall c \in M [c^{\prime} \leq c]} \lor c^{\prime}\leq 0\right]\\
\Longleftrightarrow &~\forall c^{\prime}\in \mathbb{R}\left[\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0\right]\\
\Longleftrightarrow &~\forall c^{\prime}\in (-\infty,0]\cup(0,\infty)\left[\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0\right]\\
\Longleftrightarrow &~c^{\prime}\in (-\infty,0]\cup(0,\infty) \Rightarrow (\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0)\\
\Longleftrightarrow &~c^{\prime}\in (-\infty,0]\lor c^{\prime}\in(0,\infty)\Rightarrow (\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0)\\
\Longleftrightarrow &~\begin{cases}c^{\prime}\in (-\infty,0]\Rightarrow (\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0) \quad \text{※ 恒真命題}\\ c^{\prime}\in(0,\infty) \Rightarrow (\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0)\end{cases}\\
\Longleftrightarrow &~c^{\prime}\in(0,\infty) \Rightarrow (\exists c \in M [c^{\prime} > c] \lor c^{\prime}\leq 0)\\
\Longleftrightarrow &~(c^{\prime}\in(0,\infty) \Rightarrow \exists c \in M [c^{\prime} > c]) \lor (c^{\prime}\in(0,\infty) \Rightarrow c^{\prime}\leq 0\quad\text{※ 矛盾命題})\\
\Longleftrightarrow &~c^{\prime}\in(0,\infty) \Rightarrow \exists c \in M [c^{\prime} > c]\\
\Longleftrightarrow &~\forall c^{\prime}\in(0,\infty)\exists c \in M [c^{\prime} > c]\\
\Longleftrightarrow &~\forall c^{\prime}\in(0,\infty)\exists c \in \{d(x,y)|y \in A\} [c^{\prime} > c]\\
\Longleftrightarrow &~\forall c^{\prime}\in(0,\infty)\exists y \in A [c^{\prime} > d(x,y)]\\
\Longleftrightarrow &~\forall \epsilon >0 \exists y \in A [d(x,y) < \epsilon]
\end{align*}
証明終