I'm reading the proof of Riemann Rearrangement Theorem in T. Tao's Analysis 1 textbook which can be found here Rearrangement Thm (the parts missing from the textbook, left as exercises for the reader, are completed by the user asking the question) but I don't understand the last line of the proof where the user says "If $u_i <l_i$ then for all $u_i \leq k\leq l_i$ we therefore have..."; to affirm that $S_k \to L$ shouldn't one prove that it is always $S_{l_i}\leq S_{k}\leq S_{u_i}$ to be able to invoke the Squeeze Theorem? I don't understand how that proof accomplishes this.

Could someone explain this part of that proof or show me another way to finish the proof?

Best regards,

lorenzo.