Sentence examples for as given in the proof from inspiring English sources

where α, β, α ′, and β ′ are as given in the proof below.

In Theorem 3.2, equality holds in both inequalities if and only if (or, equivalently, ), as given in the proof of Theorem 3.1.

Construct two sequences and such that and for all and and for some, as given in the proof of Theorem 2.1.

In fact, by the similar method as given in the proof of Lemma 2.5 in [1], we can prove that the function Γ satisfies the conditions (A1), (A3), and (A4).

Remark 3.6 By the same way as given in the proof of Theorem 3.3, we can prove that if the condition " {x n } is bounded" in Theorem 3.5 is replaced by the condition "if there exist constants K, K* > 0 such that ζ(t) ≤ K*t, ∀t ≥ K", then the conclusions of Theorem 3.5 still hold.

In fact, by the same methods as given in the proofs of (3.13), (3.20) and (3.29), we can prove that, and (as ) for each.

Moreover, by the same method as give in the proof of Theorem 3.1 we can also prove that the limit lim n → ∞ ∥ x n − p ∥ exists for each p ∈ F and lim n → ∞ d ( x n, F ) = 0. Therefore, we can choose a subsequence { x n k } of { x n } and a sequence { p k } in ℱ such that for any positive integer k ∥ x n k − p k ∥ < 1 2 k.

Consequently, using the same proof procedures as given in the corresponding part of the proof of Theorem 3.1 and taking into account Lemma 4.1, we can show the boundedness of as.

Proof With the same arguments as given in the first portion of the proof of Theorem 3.3, we know Γ : W 0 → 2 W 0 is a bounded map with convex values and is closed on W 0. Now, we prove the values of Γ are compact in C ( [ 0, 1 ] ; X ).

Now, following exactly the same arguments as those given in the proof of Theorem 15, we obtain the following facts: (1) For all n ∈ N, p ( x n, x n + 1 ) ≤ α ¯ ( x n − 1, x n ) p ( x n − 1, x n ),  .

Consider the construction given in the proof of Theorem 9.