Show that there is either an increasing sequence or a decreasing sequence of points $x_n$ in A with $lim_{n\rightarrow \infty} x_n=a$.

Let a be a set accumulation point $A$ subset of $\mathbb{R}$. Show that there is either an increasing sequence or a decreasing sequence of points $x_n$ in A with $lim_{n\rightarrow \infty} x_n=a$.

Could I have an answer in detail (especially the build of that sequences), please?

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  • Could you rewrite the proof, just in the first case, and build xn1, xn2 and xn3 explicitly, please? (Specially, I didn't get how does 1/n(k+1)<|xn(k+1)-a| works )

The answer is accepted.