My course notes (mathematics BSc, second-year module in real analysis, unpublished) have a proof of the Boundedness Theorem which begins:
But does that sequence work? Here's my reasoning.
Let \begin{aligned} & D=[0,2], \\ & f:D\to\mathbb{R} \text{ be given by } f(x)=x \; \forall \; x \in D; \end{aligned} then \begin{aligned} \inf\{x\in D:f(x)>1\} & = \lim_{h\rightarrow0}(1+h) \\ & = 1 \end{aligned} but \begin{aligned} |f(1)| & = 1 \\ & \ngtr 1 \end{aligned} so the proposed sequence $(x_n)$ need not give a function value with the desired property $|f(x_n)|>n \; \forall \; n \in \mathbb{N}$ .
Am I right?
Writing my comment as an answer, so this question can be deemed "answered".
Your example is correct, the flaw in the proof can be fixed by replacing $n$ by $n + 1$ in the definition of $x_n$.