Bertsekas' book on non linear programming says $y_m = sup \{x_k | k >= m\}$ has a limit whenever the sequence $ {x_k}$ is bounded above. His reasoning is that this sequence is always monotonically non-increasing.
Consider the sequence $x_k = 2-k$. Its bounded above but not bounded below. And $y_m = sup \{x_k | k >= m\} = x_m$. So this sequence does not converge even though its bounded above
Is bertsekas wrong or have I misunderstood something