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?
Answer
Answers can be viewed only if
 The questioner was satisfied and accepted the answer, or
 The answer was disputed, but the judge evaluated it as 100% correct.
Mathe
2.9K

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.
Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.
 answered
 320 views
 $5.00
Related Questions
 Assume there is no $x ∈ R$ such that $f(x) = f'(x) = 0$. Show that $$S =\{x: 0≤x≤1,f(x)=0\}$$ is finite.
 Need Upper Bound of an Integral
 For each A ∈ { Z, Q, } find the cardinality of the set of all increasing bijective functions f : A → A.
 $\textbf{I would like a proof in detail of the following question.}$
 real analysis
 Suppose that $T \in L(V,W)$. Prove that if Img$(T)$ is dense in $W$ then $T^*$ is onetoone.
 Find $\lim\limits _{n\rightarrow \infty} n^2 \prod\limits_{k=1}^{n} (\frac{1}{k^2}+\frac{1}{n^2})^{\frac{1}{n}}$
 Prove the uniqueness of a sequence using a norm inequality.