Equivalent statement on Cauchy sequence

Question

There are three statements that I need to show their equivalence:

(i) Every Cauchy sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$.

(ii) Every monotone and bounded sequence in $\mathbb{R}$ converges to a limit in $\mathbb{R}$.

(iii) Every bounded sequence in $\mathbb{R}$ has a sub-sequence that converges to a limit in $\mathbb{R}$.

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What I have thought of so far is:

1. Every Cauchy sequence in $\mathbb{R}$ is bounded. Thus, a limit exists in $\mathbb{R}$.

2. Because it is bounded, it has a sub-sequence using Monotone Subsequence Theorem.

3. Then, if a sub-sequence of a Cauchy sequence converges to a limit in $\mathbb{R}$, it implies that the Cauchy sequence itself also converges to the same limit.

In the above draft, are there any flaws? In fact, I read some materials before I figure this relation. Is it enough to show these three statements are equivalent? If not, what did I miss??