Reading the completion theorem, there is a step concerned to prove a set inclusion.
Let $\phi:A\rightarrow \tilde{A}$ be a mapping. The item $(2)$ of the proof shows that $\phi(x)=\phi(y) \implies x=y$. I do not understand why it is equivalent to say $A\subseteq \tilde{A}$. Can you clarify this for me? Has it anything to do with injectivity?
Thanks in advance!
It's abuse of notation, really, since $A$ and $\tilde{A}$ contain objects of a different nature: $A$ consists of points in the original metric space, while $\tilde{A}$ consists of equivalence classes of Cauchy sequences of points in $A$. So you're right that it can't be set inclusion what they're trying to prove.
What they really mean (they allude to this just after the proof) is that $\tilde{A}$ contains an isomorphic copy of $A$, i.e., that $\phi$ is a bijection from $A$ to $\phi(A) \subseteq \tilde{A}$. To this end, they show, as you stated, that $\phi$ is injective, showing that $A$ can be embedded in $\tilde{A}$.
I have to say, I'm a bit confused as to why the article states later on that they're going to show that $\phi$ is injective. They already did.