Let \( S \) be a subset of \( \mathbb{R} \) such that 2018 is an interior point of \( S \). Which of the following is (are) TRUE?
Show Hint
- \( S \) contains an interval
- There is a sequence in \( S \) which does not converge to 2018
- There is an element \( y \in S \), \( y \neq 2018 \), such that \( y \) is also an interior point of \( S \)
- There is a point \( z \in S \), such that \( |z - 2018| = 0.002018 \)
The Correct Option is A, B, C
Solution and Explanation
Since 2018 is an interior point of (S), there exists \(\varepsilon>0\) such that
\((2018-\varepsilon, 2018+\varepsilon)\subset S\).
(A) True: (S) contains an open interval.
(B) True: Pick any constant sequence \(x_n = 2018+\varepsilon/2 \in S\); it does not converge to 2018.
(C) True: Any \(y \neq 2018\) in \((2018-\varepsilon, 2018+\varepsilon)\) is also an interior point.
(D) False: The interval need not be wide enough to include distance (0.002018).
Answer: A, B, C
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