How to prove that every uniformly continuous set is a closed set.

Question

According to the classroom notes "Uniformly Continuous Linear Set" in American Mathematical Monthly, Vol. 62. No. 8(Oct., 1955) pp. 579-580, Author: Norman Levine link.

I'm usually confused about the state "without loss of generality".

Let see the proof of theorem 1. From the proof of theorem 1, how does "WLOG" work?

Thanks for all answers.

Answer 1

Hint: Uniformly continuous functions can be extended continuously to their completions. If $p$ is a limit point of $X$, consider the function

$$f(x)=\frac{1}{d(x,p)}$$

Answer 2

Perhaps you need the statement:

If a convergent sequence is not eventually constant, then it has a strictly monotone subsequence.

You read it in$\,$ C.G.Denlinger Elements of Real Analysis $\,$pag. 113.

Addendum (referring to my comment to prove the not uniformly continuity of $f$) the convergent sequence is a Cauchy sequence; therefore there exist always two terms $p_r$ and $p_s$, whatever near, such that $|f(p_r)-f(p_s)|=1$. Take $\epsilon<1$ ...