Using the Compactness Theorem to Show Distance Between c and d

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How to write down the sentences saying that c and d are far apart?

And what's the purpose to do so for showing that there is an elementarily equivalent structure B that is not connected.

Actually this is a question from the logic textbook " A mathematical introduction to logic " by Herbert B. Enderton.

I post this question here since it relates to pure logic and I hope someone can give me any advice.

Thank you.

Sincerely
 
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For each n, let phi_n be the sentence saying that the distance between c and d is at least n. Now try applying the compactness theorem to this set of sentences.