Sentence examples for reasoning of the proof from inspiring English sources

By the same reasoning of the proof of Theorem 2.3, we have d ( f, F ) ≤ 1 1 - L d ( f, T f ).

The proof of Lemma 8 follows the same reasoning of the proof of Proposition 2. Here, ∀n ∈ {1, …, 8}, A n ′ = lim SNR → ∞ A n, where the sets A 1, …, A 8 are given by Proposition 2. In the following lemma, we describe the set of NE of the game G ( b ) in the high SNR regime.

By using a similar reasoning to the proof of Theorem 2.9, we see that the sequence ({x_{n}}) is a w-Cauchy sequence and thus, by w-completeness, there exists (x^in X_{w}) such that (x_{n}rightarrow x^ ) as (n rightarrow infty).

Following the reasoning in the proof of Theorem 2.4 and using instead of, we deduce the conclusion of Theorem 2.6.

Proof Following the reasoning in the proof of Theorem 3.1, we use ℱ instead of F ∩ B r 0 ( x 1 ).

Next, following the reasoning in the proof of Theorem 3.1 and using ℱ instead of F ∩ B r 0 ( x 1 ), we deduce the conclusion of Theorem 3.9.

By repeating mutatis mutandis the reasoning in the proof of Proposition 4.3, one shows that the restriction of (F_{g}) to ([0,1)) also consists of four pieces; two rigid rotations interspersed by a contracting and an expanding piece.

Next, following the reasoning in the proof of Theorem 3.1 and using F(S) ∩ Ω instead of B r 0 ( x 1 ) ∩ F ( S ) ∩ Ω, we deduce the conclusion of Theorem 3.8.

By the same reasoning as the proof of Theorem 2.1, it is quadratic.

The reasoning in the proof of Theorem 2.2 shows that Δ is weakly closed, so Δ is weakly compact.

(2.12) The reasoning in the proof of Theorem 2.2 yields that (N= bigcap_{Min L} M) is weakly compact.