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**particlejohn** If $\displaystyle a_n $ is a sequence such that $\displaystyle \frac{a_{n}-1}{a_{n}+1} \to 0 $, then does $\displaystyle \lim_{n \to \infty} a_{n} $ exist? So in other words, does $\displaystyle a_n \to L $?

Assume that $\displaystyle s_n = \frac{a_{n}-1}{a_{n}+1} $. Let $\displaystyle \varepsilon > 0 $. Then there exists an $\displaystyle N \in \mathbb{N} $, such that whenever $\displaystyle n \geq N $, then $\displaystyle |s_{n}| < \varepsilon $.

Now the problem is to express $\displaystyle a_{n} $ in terms of $\displaystyle s_{n} $ and use the definition of convergence to see if it converges or not?