That's correct. Weak inequalities are preserved by sup's (as are inf's and limits). But strict inequalities are not necessarily preserved by sup's (or by inf's or limits).
That's correct. Weak inequalities are preserved by sup's (as are inf's and limits). But strict inequalities are not necessarily preserved by sup's (or by inf's or limits).
Okay, thanks ^^
Is there a proof of that ? And an example for strict inequalities ?
Is there a proof of that ? And an example for strict inequalities ?
Proof by contradiction: suppose that for all n. If then there must exist an x_n arbitrarily close to s. In particular, it can be made greater than a ...
Counterexample for strict inequalities: . But , which is not strictly less than 1.