In the proof of Theorem 7.11 of Baby Rudin, it is mentioned the sequence $\{ A_n \} _n$ is convergent because it is Cauchy. I didn't initially understand this as $E$, the domain of the functions was not declared to be complete. The above answer in the link tells that since they belong to $\mathbb{R}$, hence they are convergent iff Cauchy as $\mathbb{R}$ is complete.
But, the limit of a function is defined to be an element belonging to the range of the function. Here we aren't told whether the range is $\mathbb{R}$.
Can someone give me the reasoning. Thanks.
You are wrong when you assume that the limit of a function is defined to be an element belonging to the range of the function. For instance, the range of the exponential function is $(0,\infty)$, but $\lim_{x\to-\infty}e^x=0\notin(0,\infty)$.