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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...

I am focused on Chapter 3: Convergent Sequences

I need some help to fully understand the proof of Theorem 3.1.1 ...Garling's statement and proof of Theorem 3.1.1 (together with some interesting remarks) reads as follows:

View attachment 9026

View attachment 9027

In the above text by Garling, in the proof of statement (iii), we read the following:

" ... ... Let \(\displaystyle A = \{ k \in \mathbb{Z}^{ + } \ : \ k \leq nx \}\). A is non-empty ( \(\displaystyle 0 \in A\) ) and finite, by (ii) ... ... " My question is as follows:

Although it is plausible (maybe even "obvious" ... ) that A is finite ... how do we logically and rigorously prove this using (ii} ... that is, logically and rigorously, how does (ii) imply that A is finite ...?

Help will be appreciated ... ...

Peter

I am focused on Chapter 3: Convergent Sequences

I need some help to fully understand the proof of Theorem 3.1.1 ...Garling's statement and proof of Theorem 3.1.1 (together with some interesting remarks) reads as follows:

View attachment 9026

View attachment 9027

In the above text by Garling, in the proof of statement (iii), we read the following:

" ... ... Let \(\displaystyle A = \{ k \in \mathbb{Z}^{ + } \ : \ k \leq nx \}\). A is non-empty ( \(\displaystyle 0 \in A\) ) and finite, by (ii) ... ... " My question is as follows:

Although it is plausible (maybe even "obvious" ... ) that A is finite ... how do we logically and rigorously prove this using (ii} ... that is, logically and rigorously, how does (ii) imply that A is finite ...?

Help will be appreciated ... ...

Peter