Doubt in Riemann Rearrangement theorem

Question

I was trying to understand proof of Rearrangement theorem.

I think I understood the proof. Only one small confusion is there

I don't know why author considered both case of $N_1,N_2$

As I thought both condition are same .

Please Help me what is I am thinking wrong

Answer 1

You are right, the first condition is not really necessary, as $$ |s_n-s_m|\le\sum_{k=m+1}^n|a_k|<\frac{\epsilon}2 $$ for $m,n\ge N_2$ already implies $$ |s_n-A|=\lim_{m\to\infty}|s_n-s_m|\le\fracϵ2 $$ which is sufficient to proceed with the proof in the same way.

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Thus first bounding the convergence then the Cauchy property is done more for optics, the narrative flow of the proof, to avoid arguments that are only tangential to the proof for the convenience of the reader; than for necessity.