why do we need to prove that the limit of a contraction is a fixed point in the contraction mapping theorem

Question

In this chapter they first prove that iteratively applying a contraction is a cauchy sequence. Since the metric space is complete we know that sequence converges, and it does so to a unique point. So why do they then additionally prove that the point of convergence is a fixed point of the contraction, isn't this already implied by the proving the sequence is cauchy. I feel like it's more of a corollary since it applies to any arbitrary function that is continuous in the domain.

Answer 1

So why do they then additionally prove that the point of convergence is a fixed point of the contraction

Because that's what the theorem states, and therefore, that's what the proof should conclude with.

isn't this already implied by the original proof.

What do you mean by "original proof"? There is only one proof and it can be divided into the following parts:

1. Prove that the sequence, defined by $x_{n+1} = T(x_n)$ converges.
2. Prove that if a sequence, defined by $x_{n+1}=T(x_n)$, converges, then the limit of the sequence is a fixed point of $T$.
3. Conclude from 1 and 2 that $T$ has a fixed point.
4. Prove that $T$ can have at most one fixed point.
5. Conclude from 3 and 4 that $T$ has exactly one fixed point.

Now, sure, alternatively, the authors could cite point number 2 above as a separate lemma, and prove it before proving the main theorem, but I'd say it's cleaner this way, for two reasons:

• The proof is short, so splitting it into subtheorems would not help readability
• The lemma itself is not all that interesting.

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Edit:

It is, of course, also possible to prove point 3 directly from 1 and from the fact that $T$ is continuous. However, this would most probably lead to a proof that is harder to follow, and possibly even longer than original.

But the mere fact that you can prove something without a particular sub-lemma does not mean that that particular lemma is "useless" for the proof. Technically, every proof in mathematics can be trimmed down to base principles, but at a great cost of readability.