MHB Coordinate-Wise Convergence in R^n .... TB&B Chapter 11, Section 11.4 ....

[Math Amateur]

I am reading the book "Elementary Real Analysis" (Second Edition, 2008) Volume II by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner ... and am currently focused on Chapter 11, The Euclidean Spaces $$\mathbb{R}^n$$ ... ...

I need with the proof of Theorem 11.15 on coordinate-wise convergence of sequences in $$\mathbb{R}^n$$ ... note that the proof precedes the theorem statement in TB&B ... ...

Theorem 11.15 and its proof (as preceding notes) reads as follows:View attachment 7711
View attachment 7712Can someone please explain exactly how (8) follows from (7) ...Help will be much appreciated ...

Peter

 

[GJA]

Hi, Peter.

Define the vector $y=x_{k}-x$, then apply $(7)$ to $y$ to get $(8)$.