I just wrote a proof to show the following statement:
Every Cauchy sequence in R converges iff Every absolutely convergent series in R is convergent.
Please let me know what you think about the proof and don't hesitate to correct me!
Proof:
First, we show that Every Cauchy sequence in $\mathbb{R}$ converges $\implies$ Every absolutely convergent series in $\mathbb{R}$ is convergent.
Let $\sum_{n=0}^\infty a_n$ be an absolutely convergent series. According to a theorem (All Cauchy sequences in complete fields are convergent), it follows that the sequence of partial sums $(S_n)_{n=0}^\infty$ also converges. That is, we have:
$$\forall \varepsilon > 0 \; \exists N \in \mathbb{N} \; \forall n \geq m \geq N: |S_n - S_m| = \left| \sum_{k=0}^n |a_k| - \sum^m_{k=0}|a_k| \right| = \left|\sum^n_{k=m+1}|a_k| \right| < \varepsilon.$$
Using the triangle inequality, we get:
$$\left| \sum^n_{k=m+1} a_k\right| \leq \sum^n_{k=m+1} |a_k| < \varepsilon,$$
which implies that the series $\sum^\infty_{n=0} a_k$ is also Cauchy and, consequently, convergent.
Now, we want to prove the converse:
Every absolutely convergent series in $\mathbb{R}$ is convergent $\implies$ Every Cauchy sequence in $\mathbb{R}$ converges.
Let $(a_k)_{k=0}^\infty$ be a Cauchy sequence, and $\sum_{n=0}^\infty a_n$ an (absolutely) convergent series. Using the Cauchy criterion:
$$\forall \varepsilon > 0 \; \exists N \in \mathbb{N} \; \forall n \geq m \geq N: \left| \sum^n_{k=m+1} a_k \right| < \varepsilon,$$
we find a representation of the Cauchy sequence as a series. Further, with the help of the partial sums of $\sum_{n=0}^\infty a_n$:
[ \begin{align*} \lim_{n\to \infty} \sum^n_{k=m+1} a_k &= \lim _{n\to \infty} \sum^n_{k=0} a_k - \lim_{m\to \infty} \sum^m_{k=0} a_k = A - A\\ &\iff\\ \lim_{n\to \infty} \sum^n_{k=0} a_k - \not A &=A - \not A\\ &\iff\\ \lim_{n\to \infty}\sum^n_{k=0} a_k &= A, \end{align*} ]
as desired. $\Box$
EDIT: Here is the updated proof:
$\textbf{Proof.}$ First, we demonstrate that $$ \text{Every Cauchy sequence in}\; \mathbb{R}\;\text{converges}\implies\text{Every absolutely convergent series in}\; \mathbb{R} \;\text{is convergent.} $$ Let $\sum_{n=0}^\infty a_n$ be an absolutely convergent series, meaning there exists a limit $A \in \mathbb{R}$ for the sequence of partial sums $(S_n)_{n=0}^\infty$ of the series, rendering the sequence $(S_n)_{n=0}^\infty$ convergent. Since all convergent sequences are Cauchy, we have: $$ \forall \varepsilon > 0\; \exists N \in \mathbb{N} \; \forall n\geq m\geq N: |S_n - S_m| = \left| \sum_{k=0}^n |a_k| - \sum^m_{k=0}|a_k| \right| = \left|\sum^n_{k=m+1}|a_k| \right|< \varepsilon. $$ Applying the triangle inequality, we obtain: $$ \left| \sum^n_{k=m+1} a_k\right| \leq \sum^n_{k=m+1} |a_k| < \varepsilon, $$ demonstrating that the series $\sum^\infty_{n=0} a_k$ is also Cauchy and, therefore, convergent.
Now, let's prove the converse: $$ \text{Every absolutely convergent series in}\; \mathbb{R} \;\text{is convergent}\implies\text{Every Cauchy sequence in}\; \mathbb{R}\;\text{converges}. $$ So, let $(a_n)_{n=0}^\infty$ be a Cauchy sequence. We construct a series $\sum^\infty_{n=0}b_n,$ where the sequence of its partial sums $(s_n)_{n=0}^\infty$ is intended to be equal to our Cauchy sequence. By definition, we have: $$ \begin{align*} s_0&:=a_0 = b_0\\ s_1&:=a_1 = b_0 + b_1\\ \vdots \\ s_n&:=a_n= b_0 + \dots + b_n. \end{align*} $$ Since $(a_n){n=0}^\infty$ is Cauchy, we also have: $$ \forall \varepsilon >0 ;\exists N \in \mathbb{N} ; \forall n\geq m\geq N: |a_n -a_m| = \left| \sum^n{k=0} b_k - \sum^m_{k=0}b_k\right|=\left| \sum^n_{k=m+1}b_k\right| < \varepsilon. $$ For the series $\sum^\infty_{n=0}b_n$, using the triangle inequality twice, we obtain: $$ \lim_{n\to \infty} \sum_{k=0}^n b_k \leq \lim_{n\to \infty} \left| \sum_{k=0}^n b_k\right| = \lim_{n \to \infty} \left| \sum_{k=0}^m b_k + \sum^n_{k=m+1} b_k \right| \leq \lim_{n \to \infty} \left| \sum_{k=0}^m b_k \right| + \left| \sum_{k=m+1}^n b_k \right|=\left|\sum_{k=0}^m b_k\right|\leq \sum_{k=0}^m |b_k|, $$ indicating that the series converges to a finite value. Thus, we have demonstrated the absolute convergence of the series and, in particular, the convergence of the sequence of partial sums $(a_n)_{n=0}^\infty$. $\Box$
