Proving the Existence of Frontier Points for Sets in ℝ

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