Do frontier points exist between a point in E and a point in E's complement?

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3.4. Let E$\in$  R. Prove or disprove the following statements:
(i) if a$\in$E and b$\in$E$^{c}$ = ℝ\E and a < b then [a ,b] $\cap$∂E IS NOT EQUAL TO ∅.
(ii) if a$\in$E and b$\in$E$^{c}$ = ℝ\E and a < b then (a ,b) $\cap$∂E IS NOT EQUAL TO ∅.


I am really stuck I know that the frontier of a set is when a sequence in E and a sequence in E$^{c}$ converge to the same limit.

 

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I am able to show this from drawing a sketch but how can I do it without a drawing?