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Latihan Persekitaran-ε

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Soal-soal berikut merupakan soal mengenai persekitaran dari suatu titik. Untuk mengakses materi mengenai persekitaran – ε, silakan kunjungi laman berikut.

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Tunjukkan bahwa setiap persekitaran-$\varepsilon$ terbuka di $\mathbb{R}$.

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Jawaban:

Diberikan $N_\varepsilon(a)$ persekitaran-$\varepsilon$ dari $a$. Untuk menunjukkan $N_\varepsilon(a)$ terbuka akan ditunjukkan untuk sebarang $x \in N_\varepsilon(a)$ terdapat $\delta_x>0$ sehingga $N_{\delta_x}(x) \subseteq N_\varepsilon(a)$. Diambil sebarang $x \in N_\varepsilon(a)$, diperolehDiberikan $N_\varepsilon(a)$ persekitaran-$\varepsilon$ dari $a$. Untuk menunjukkan $N_\varepsilon(a)$ terbuka akan ditunjukkan untuk sebarang $x \in N_\varepsilon(a)$ terdapat $\delta_x>0$ sehingga $N_{\delta_x}(x) \subseteq N_\varepsilon(a)$. Diambil sebarang $x \in N_\varepsilon(a)$, diperoleh\[|x-a| < \varepsilon.\]Diambil $\delta_x= \frac{1}{3}\min\{\varepsilon-|x-a|, |x-a|\}$. Diperoleh $\delta >0$.\\Diambil $z \in N_{\delta_x}(x)$, maka $|x-z|< \delta_x$. Akibatnya,\[\begin{array}{lll}|z-a| &\leq |z-x|+|x-a|\\ &<\delta_x + |x-a|\\ &< (\varepsilon-|x-a|)+|x-a|\\ &< \varepsilon.\end{array}\]Berarti $z \in N_\varepsilon(a)$. Jadi, $N_\varepsilon(a)$ terbuka.

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Mari Belajar Bersama Kami!

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