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I am reading N. L. Carothers' book: "Real Analysis". ... ...
I am focused on Chapter 3: Metrics and Norms ... ...
I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following:
View attachment 9216
In the above text from Carothers we read the following:
" ... ... it follows that a sequence of vectors $$x^{ (k) } = ( x_1^k, \ ... \ ... \ , x_n^k)$$ in $$\mathbb{R}^n$$ converges (is Cauchy) if and only if each of the coordinate sequences $$( x_j^k )$$ converges in $$\mathbb{R}$$ ... ... "
My question is as follows:
Why exactly does it follow that a sequence of vectors $$x^{ (k) } = ( x_1^k, \ ... \ ... \ , x_n^k)$$ in $$\mathbb{R}^n$$ converges (is Cauchy) if and only if each of the coordinate sequences $$( x_j^k )$$ converges in $$\mathbb{R}$$ ... ... ?
Help will be appreciated ...
Peter
I am focused on Chapter 3: Metrics and Norms ... ...
I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following:
View attachment 9216
In the above text from Carothers we read the following:
" ... ... it follows that a sequence of vectors $$x^{ (k) } = ( x_1^k, \ ... \ ... \ , x_n^k)$$ in $$\mathbb{R}^n$$ converges (is Cauchy) if and only if each of the coordinate sequences $$( x_j^k )$$ converges in $$\mathbb{R}$$ ... ... "
My question is as follows:
Why exactly does it follow that a sequence of vectors $$x^{ (k) } = ( x_1^k, \ ... \ ... \ , x_n^k)$$ in $$\mathbb{R}^n$$ converges (is Cauchy) if and only if each of the coordinate sequences $$( x_j^k )$$ converges in $$\mathbb{R}$$ ... ... ?
Help will be appreciated ...
Peter