Now let (theta ll c) be arbitrary in (mathcal {A}).
For each (a,b,cin E) such that (apreceq b ll c), we have (all c).
If (theta preceq ull c) for each (theta ll c), then (u=theta). .
(i) ({p_{n}}) converges to p if, for each (cin {mathcal{A}}) with (theta ll c), there exists (n_{0}in mathbb{N}) such that (d_{mathrm {lc}}(p_{n},p) ll c) for all (nge n_{0}).
For all (cin E) with (theta ll c), if there exists a positive integer N such that (d(x_{n},x_{m})ll c) for all (n,m>N), then ({x_{n}}) is called a Cauchy sequence in X. 3.
Now, by Lemma 4, for any (cin E) with (theta ll c) there exists a positive integer (N_{1}) such that (d z, y_{n})ll c) for all (ngeq N_{1}).
If E is a real Banach space with a solid cone P and if (thetapreccurlyeq u ll c) for each (thetall c), then (u = theta).
For all (cin E) with (theta ll c), if there exists a positive integer N such that (d(x_{n},x ll c) for all (n>N), then ({x_{n}}) is said to be convergent and x is the limit of ({x_{n}}).
If (mathcal{A}) is a real Banach space with a solid cone P and if (|x_{n} |rightarrow0) ((nrightarrowinfty)), then for any (theta ll c), there exists (Ninmathbb{N}) such that, for any (n > N), we have (x_{n}ll c).
(ii) ({p_{n}}) is a Cauchy sequence if and only if for each (cin {mathcal{A}}) with (theta ll c), there exists (n_{0}in mathbb{N}) such that (d_{mathrm {lc}}(p_{n},p_{m}) ll c) for all (n,mge n_{0}).
({p_{n}}) is a Cauchy sequence if and only if for each (cin {mathcal{A}}) with (theta ll c), there exists (n_{0}in mathbb{N}) such that (d_{mathrm {lc}}(p_{n},p_{m}) ll c) for all (n,mge n_{0}).
