Sentence examples for achieves the proof from inspiring English sources

This achieves the proof.

This achieves the Proof of Theorem 2.

Finally, the dominated convergence theorem achieves the proof.

Now the use of Theorem 3.1 achieves the proof.

Applying Theorem 3.3, this achieves the proof of the existence of mild solutions of the system (4.1 - 4.3 4.1 - 4.3

Choosing δ conveniently and applying the discrete Gronwall inequality, we achieve the proof of Lemma 7. ■.

If (n-k) is odd, we can do similar arguments to achieve the proof.

Now, to achieve the proof of (5.10), we would like to replace the (E_j) with a sequence of sets converging to B in (C^1) and contradicting inequality (5.8).

Thus, to achieve the proof of (4.33) we only need to show that begin{aligned} int _I|tau ^prime (t -1|,dt -1|(n)sqrt{D(E^*)}.

Hence gammabigl overline{co}(A bigr) (t) leqgamma(A) (t),quad 0 leq t leq T. It follows that (gamma overline{co}(A))leqgamma(A)), by which we achieve the proof for the first assertion.

Since (xi_{n}) is a Cauchy sequence in (X_{T}) and contains a subsequence (xi_{m_{k}}) which converges to ξ, (xi_{n}) converges to ξ in (X_{T}), and by this we achieve the proof.