Suppose limn→∞ (Sn − 1)/ (Sn + 1) = 0. Prove that limn→∞ Sn = 1.
Given that limn→∞Sn−1Sn+1=0lim_{n \to \infty} \frac {S_n-1}{S_n+1}=0limn→∞Sn+1Sn−1=0
⟹ limn→∞(Sn−1)limn→∞(Sn+1)=0\implies \frac {lim_{n \to \infty} (S_n-1)}{lim_{n \to \infty}( S_n+1)}=0⟹limn→∞(Sn+1)limn→∞(Sn−1)=0 ........(1)
Now from (1) we can say that as the exists , therefore limn→∞(Sn+1)≠0lim_{n \to \infty}( S_n+1) \neq 0limn→∞(Sn+1)=0
So the only possibility is that , limn→∞(Sn−1)=0lim_{n \to \infty} (S_n-1)=0limn→∞(Sn−1)=0
⟹ limn→∞Sn=1\implies lim_{n \to \infty} S_n=1⟹limn→∞Sn=1
Which completes the proof.
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