Carothers (p. 90-91) includes the following lemma and theorem when describing the sequential criterion for total boundedness. I'm having trouble understanding how they are different. Could someone please explain if possible? (More below).
Lemma 7.3 Let $(x_n)$ be a sequence in $(M, d)$ and let $A = \{x_n: n\geq 1\}$ be its range:
(i) If $(x_n)$ is Cauchy then $A$ is totally bounded
(ii) If $A$ is totally bounded then $(x_n)$ has a Cauchy subsequence
Later we have:
Theorem 7.5 A set $A$ is totally bounded if and only if every sequence in $A$ has a Cauchy subsequence.
The proof of Theorem 7.5 states that "the forward implication is clear from Lemma 7.3" and then uses the contrapositive to prove that if every sequence in $A$ has a Cauchy subsequence then $A$ is totally bounded.
Why they look the same to me: If $A$ is totally bounded then by Lemma 7.3 (ii) an arbitrary sequence in it has a Cauchy subsequence; and if every sequence in $A$ has a Cauchy subsequence, then because the subsequence is also in $A$, by Lemma 7.3 (i), $A$ is totally bounded.
Am I missing something?
Your last part is not correct. If every sequence in $A$ has a Cauchy subsequence you have proved that $\{x_n: n\geq 1\}$ is totally bounded for any sequence $(x_n)$ in $A$ , not that $A$ itself is totally bounded. Note that $A$ is a countable set in Lemma 7.3 so it does not directly give you total boundedness of $A$ in Theorem 7.5 (because $A$ may be uncountable).