Sentence examples for bounded by using the from inspiring English sources

where the second term can be upper bounded by using the th dominant sample.

Since for each i ∈ N, ( x n ( i ) ) i = 1 ∞ is bounded, by using the diagonal method, we can find a subsequence ( x n j ) of ( x n ) such that ( x n j ( i ) ) converges for each i ∈ N with 1 ≤ i ≤ l.

Next, other right terms of (12) are shown to be bounded by using the Cauchy-Schwarz inequality, the continuity of injection of (mathbf{H}^{mathbf{1}}(boldsymbol{Omega})) in (mathbf{L}^{infty}(boldsymbol{Omega})) and by taking into account assumptions (3) on the function g.

Let and with for Take There exists such that Let Since for each is bounded, by using the diagonal method, we have that for each, we can find a subsequence of such that converges for all with Since is Cauchy sequence for all there exists such that (4.7).

end{aligned} (48) The second term of (47) can be shown to be bounded, by using (3), the Cauchy-Schwarz inequality and Lemma 2.

The upper bound can be derived following the same steps as Appendix B. It turns out that at high SNR, the gap rate is upper bounded by, Using Lemma 2, the expected gap rate is upper bounded by, (45).

The uncertainties in the system are bounded by using a frequency-dependent function.

where the terms on the right-hand side can be bounded by using Lemmas 2.1 2.3 as follows: (2.18).

All of these terms, except the last one, can be bounded by using inequalities already proved.

Similarly, we can prove that H is bounded below by using the method described above.

It is not difficult to show that is compact for any open bounded set by using the Arzela-Ascoli theorem.