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Let $H$ be a Hilbert space and $C$ a convex subset. The Hilbert space projection theorem states that there exists a unique $c_0\in C$ with $\|c_0\| = \inf_{c \in C}\|c\|$.

I'm confused about the proof. By the properties the infimum satisfies, for every $n \in \mathbb N$ there exists $c_n \in C$ such that $$ \|c_n\| \le \inf_{c\in C}\|c\| + {1\over n}$$

If we let $c_n$ denote the resulting sequence then it seems to me that this sequence converges to $\inf_{c\in C}\|c\|$. Since it converges it must be a Cauchy sequence.

But in the proof e.g. here it is shown that this sequence is a Cauchy sequence.

Is it wrong to think that $c_n$ converges and therefore is a Cauchy sequence?

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You are correct that $\|c_n\|$ converges and is thus Cauchy. That is insufficient however to conclude that $c_n$ is Cauchy.

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