Sentence examples for bounded and the sequence from inspiring English sources

According to (3.7), the sequence ({u^{ k)}}) is bounded and the sequence ({|delta_{u}^{ k)}|_{Gamma_{S}}}) is monotonically decreasing.

(32) Since the sequences ({mathbf{f}_{epsilon_{k}}}_{k=1}^{infty}) and ({mathbf{g}_{epsilon_{k}}}_{k=1}^{infty}) are uniformly essentially bounded on ([0,T]), from (4), we see that the sequence ({beta^_{epsilon_{k}}}_{k=1}^{infty}) is bounded, and the sequence ({mathbf{z}^_{epsilon_{k}}}_{k=1}^{infty}) is uniformly essentially bounded on ([0,T]).

This implies that the sequence { ∥ x n − p ∥ } n ∈ N is bounded and hence the sequence { x n } n ∈ N is bounded.

On the other hand, noticing that the sequence is bounded and the duality mapping is single-valued and norm to weak* uniformly continuous on bounded subsets of, we conclude that (4.11).

From the proof of Theorem 1.3, we need to prove that any ((PS _{c}) sequence is bounded and the condition (A2) is satisfied.

end{aligned} Thus the sequence ({ Vert x_{k}-x^ Vert }) is decreasing and convergent, and hence the sequence ({x_{k}}) is bounded, and from (2.9), the sequence ({d_{k}}) is also bounded.

Therefore, the sequence is bounded, and so are the sequences,,, and,.

Consider the subsequences { a σ 1 ( m ) 2 } m ∈ N, { a σ 1 ( m ) 3 } m ∈ N, …, { a σ 1 ( m ) n } m ∈ N, that are n − 1 real bounded sequences, and the sequence { b σ 1 ( m ) } m ∈ N that also converges to δ.

This procedure converges to the local minima because of the fact that the objective function is bounded below, and the sequence of function values is monotonically decreasing, and the gradients at the convergence are zeros.

This implies that the sequence { d ( x n, p ) } is bounded, and so is the sequence { x n }.

This implies that is bounded and so is the sequence.