Sentence examples for a strong convergent from inspiring English sources

The sequence { u m } has a strong convergent subsequence.

Hence, ({u_{n}}) possesses a strong convergent subsequence in X.

Claim 1. { u n k } n = 1 ∞ ⊂ E possesses a strong convergent subsequence in E, a.e.

Finally, we show that ({u_{n}}) possesses a strong convergent subsequence.

Then any bounded ((mathrm{PS})_{c}) sequence of J has a strong convergent subsequence.

So ({u_{n}}) is bounded in H. 2. We prove that ({u_{n}}) has a strong convergent subsequence.

In fact, for any strong convergent sequence { x n } ⊂ X such that x n → x 0 and ∥ x n − T n x n ∥ → 0 as n → ∞, there must be x 0 = 0 ∈ ⋂ n = 1 ∞ F ( T n ).

On the other hand, for any strong convergent sequence ({z_{n}}subset E) such that (z_{n}rightarrow z_{0}) and (|z_{n}-T_{n}z_{n}|rightarrow0) as (nrightarrowinfty), it is easy to see that there exists a sufficiently large nature number N such that (z_{n}neq x_{m}) for any (n, m >N).

In fact, for any strong convergent sequence ({z_{n}}subset E) such that (z_{n}rightarrow z_{0}) and (|z_{n}-T_{n}z_{n}|rightarrow0) as (nrightarrowinfty), there exists sufficiently large natural number N such that (z_{n}neq x_{m}), for any (n, m >N).

Proof In fact, for any strong convergent sequence { z n } ⊂ E such that z n → z 0 and ∥ z n − T z n ∥ → 0 as n → ∞, from Conclusion 3.9, there exists a sufficiently large natural number N such that z n ≠ x m, for any n, m > N. Then T z n = − z n for n > N, it follows from ∥ z n − T z n ∥ → 0 that 2 z n → 0 and hence z n → z 0 = 0. □.

Proof In fact, for any strong convergent sequence { z n } ⊂ E such that z n → z 0 and ∥ z n − T n z n ∥ → 0 as n → ∞, from Conclusion 3.2, there exists a sufficiently large natural number N such that z n ≠ x m for any n, m > N. Then T z n = − z n for n > N, it follows from ∥ z n − T n z n ∥ → 0 that 2 z n → 0 and hence z n → z 0 = 0. □.