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I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##. The way I understood this part of the proof is, he has already shown that ##f(b)=c## when he chose an

**arbitrary**sequence at the part where I underlined with red. So why is he repeating the argument after that?