Let $(X, ||\cdot||)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X. Show that the seqience Show that the sequence $λ_n = ||x_n − y_n|| $ converges.
Answer
Let $\epsilon>0$, and choose $N_0$ large enough so that for all $n,m\geq N_0$
\[||x_n-x_m||< \epsilon, ||y_n-y_m||<\epsilon/2.\]
By the reverse triangle inequality we have
\[|\lambda_n-\lambda_m|=| ||x_n-y_n|| - ||x_m-y_m||| \leq ||x_n-y_n - (x_m-y_m)|| \]
\[=||x_n-x_m - (y_n-y_m)|| \leq ||x_n-x_m||+ || y_n-y_m|| \]
\[<\epsilon/2+\epsilon/2=\epsilon.\]
Hence for all $n,m \geq N_0$ we have
\[|\lambda_n-\lambda_m| < \epsilon,\]
i.e. the sequene $\{\lambda_n\}$ is Cauchy, and thus it is a convergent series.

4.8K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1399 views
- $15.00
Related Questions
- Define$ F : C[0, 1] → C[0, 1] by F(f) = f^2$. For each $p, q ∈ \{1, 2, ∞\}$, determine whether $F : (C[0, 1], d_p) → (C[0, 1], d_q)$ is continuous
- How to derive the term acting like a first derivative with respect to A that I found by accident?
- Prove the uniqueness of a sequence using a norm inequality.
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
- $\textbf{I would like a proof in detail of the following question.}$
- Analyzing the Domain and Range of the Function $f(x) = \frac{1}{1 - \sin x}$
- Prove that $A - B=A\cap B^c$
- What is the Lebesgue density of $A$ and $B$ which answers a previous question?