Sentence examples for bounded and the proof from inspiring English sources
Thus ({u_{n}}) is bounded, and the proof completed.
Since θ > p +, we infer that { x n } is bounded and the proof is complete.
(4.5) This shows that the nontrivial periodic solution of system (1.4) is uniformly bounded and the proof is complete.
end{aligned} (2.11) Since (1< m< p< q), we conclude that (|u|_{E}) is bounded and the proof is complete.
Thus, there exists (hat{C}>0) such that (| Y t,x |leqhat{C}), so that (Y t,x)) is bounded, and the proof is complete.
for some constants C ′, C ″ > 0, from which we conclude that u ˜ is bounded and the proof of Theorem 1.1 is complete.
Hence, the solutions of (3.7) are bounded in, and the proof is complete.
The original integral thus has a bounded value, and the proof is complete.
Since and the series converges, the above inequality contradicts the fact that is bounded, hence, and the proof is complete.
In view of [19], it is easy to see that the pair ((A,B) ) generates a propagation family (W t)) of uniformly bounded, and similarly to the proof of (2.15), (2.16), and (2.17) in [19], we can see that ({W t)}_{tgeq 0}) is norm-continuous for (t>0) and (Vert W t) Vert leq 1), that is, assumption (H1) is satisfied.
A minor gap is visible between the SUS simulation curve (true SUS power) and the SUS analytical upper bound as the orthogonalized norm distributions were bounded in the proof (see Appendix D for details).
