Question

Let \( S \) be a subset of \( \mathbb{R} \) such that 2018 is an interior point of \( S \). Which of the following is (are) TRUE?

• A. \( S \) contains an interval
• B. There is a sequence in \( S \) which does not converge to 2018
• C. There is an element \( y \in S \), \( y \neq 2018 \), such that \( y \) is also an interior point of \( S \)
• D. There is a point \( z \in S \), such that \( |z - 2018| = 0.002018 \)

Hint:

Interior points imply the existence of a neighborhood around them that lies entirely within the set.

Answer 1

The Correct Option is A, B, C

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