Triangle Inequality and Convergence of ${y}_{n}$

In summary, the article discusses a sequence ${y_n}$ in a metric space X that converges to ${x}^{*}$, and explores the relationship between ${y_n}$ and ${y_{m+1}}$. The sequence satisfies a specific condition and the triangle inequality is used in the proof of Theorem 3.3. The proof also shows that $\lim_{{n}\to{\infty}}{y}_{n}={x}^{*}$ by taking the limit as $n\to\infty$ of ${w}_{\lambda}\left({y}_{n},{x}^{*}\right)$.
  • #1
ozkan12
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Let ${y}_{n}$ be a arbitrary sequence in X metric space and ${y}_{m+1}$ convergent to ${x}^{*}$ in X...İn this case by using triangle inequality can we say that ${y}_{n}\to {x}^{*}$
 
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  • #2
What's the relationship between $y_n$ and $y_{m+1}$? Is it the same sequence? On the one hand, it would seem not, because $y_n$ is arbitrary. On the other hand, the notation would indicate to me that they are the same sequence. Is it a subsequence?
 
  • #4
Very helpful link, thank you!

The sequence $\{y_n\}$ is by no means arbitrary. It satisfies a very particular condition. It's not an iterative sequence, because $y_{n+1}\not\equiv T y_n$. However, the condition that the sequence satisfies ensures that $y_{n+1}$ is getting "closer and closer" - in the modular sense - to the quantity $Ty_n$ as $n\to\infty$. That's what
$$w_{\lambda}(y_{n+1},Ty_n)\le \varepsilon_n$$
guarantees.

To answer the question in the OP, there is a triangle inequality invoked in the proof of Theorem 3.3. It's used in the third line down from the word "Proof", where you have
$$w_{\frac{\lambda\cdot m}{m}}(T^{m+1}x,y_{m+1}) \le
w_{\frac{\lambda(m-1)}{m}}(T^{m+1}x,Ty_m)+w_{\frac{\lambda}{m}}(Ty_m,Ty_{m+1}).$$
 
  • #5
Dear Ackbach

İn this case, How we say that $\lim_{{n}\to{\infty}} {y}_{n}={x}^{*}$ in this article...? I found something but I am not

sure...I wrote it..

${w}_{\lambda}\left({y}_{n},{x}^{*}\right)={w}_{\frac{\lambda}{2}}\left({y}_{n},{T}^{n}x\right)+{w}_{\frac{\lambda}{2}}\left({T}^{n}x,{x}^{*}\right)$
$\le\sum_{i=0}^{n-1}{k}^{n-1-i}{\varepsilon}_{i}+{w}_{\frac{\lambda}{2}}\left({T}^{n}x,{x}^{*}\right)$

By taking limit as $n\to\infty$ we get $\lim_{{n}\to{\infty}}{w}_{\lambda}\left({y}_{n},{x}^{*}\right)$...İs this true ?

Thank you for your attention :)
 

FAQ: Triangle Inequality and Convergence of ${y}_{n}$

1. What is the Triangle Inequality?

The Triangle Inequality is a mathematical principle that states that the sum of any two sides of a triangle must be greater than the length of the third side. In other words, the shortest distance between two points is a straight line, which is always shorter than the sum of any other two sides of a triangle.

2. How is the Triangle Inequality related to convergence of ${y}_{n}$?

The Triangle Inequality is a key concept in the study of convergence of ${y}_{n}$, as it helps to determine whether a sequence is convergent or not. In order for a sequence to be convergent, it must satisfy the Triangle Inequality, meaning that the limit of the sequence must be less than or equal to the sum of the limits of the individual terms in the sequence.

3. What is the significance of the Triangle Inequality in mathematics?

The Triangle Inequality is a fundamental mathematical principle that is used in various areas of mathematics, such as geometry, calculus, and analysis. It provides a basis for proving theorems and solving problems in these fields, and is also used in real-world applications such as optimization and data analysis.

4. Can the Triangle Inequality be used to prove the convergence of ${y}_{n}$?

Yes, the Triangle Inequality can be used to prove the convergence of ${y}_{n}$. Specifically, if a sequence ${y}_{n}$ satisfies the Triangle Inequality, then it is also convergent. However, the converse is not always true, as there are sequences that may be convergent but do not satisfy the Triangle Inequality.

5. How can the Triangle Inequality be applied in practical situations?

The Triangle Inequality has many practical applications, such as in engineering, physics, and computer science. It can be used to optimize the shortest path between two points, calculate the minimum and maximum possible values for a given quantity, and determine the stability of a system. In computer science, the Triangle Inequality is used in algorithms for data compression and error correction.

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