'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in theorem proofing and in practice.
In general, a optimization algorithm starts from a $x_0$, then in the $k$ iteration and find a direction $d_k$, then line search find a suitable step size $\alpha_k$ such that $f(x_{k+1}) = f(x_k + \alpha_k d_k) \leq f(x_k)$, until reaches some termination criteria.
Is that 'The level set is bounded' is just used to ensure that $x_k$ won't go to infinity as $k \rightarrow \infty$ in the algorithm process ?
The usual point is not really that the level set is bounded, but that it is compact. So that from the continuity of the function under consideration it follows that a minimum exists. Which fact can then be used to explore the convergence of minimizing methods.
But in the usual cases, the domain is inside a finite-dimensional space, so that bounded (and closed) is equivalent to compact.