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More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
The hyperbolic bound [7] is a tighter sufficient condition for schedulability than the one presented by Liu and Layland: = (+), where U i is the CPU utilization for each task. It is the tightest upper bound that can be found using only the individual task utilization factors.
Then has an upper bound (, for example, or ) but no least upper bound in : If we suppose is the least upper bound, a contradiction is immediately deduced because between any two reals and (including and ) there exists some rational , which itself would have to be the least upper bound (if >) or a member of greater than (if <).
If (,) is a partially ordered set, such that each pair of elements in has a meet, then indeed = if and only if , since in the latter case indeed is a lower bound of , and since is the greatest lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original ...
In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers ... converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums ...
Proof It is clear that the sequence of polynomials of best approximation p n ∗ ( x ) {\displaystyle p_{n}^{*}(x)} converges to f ( x ) uniformly (due to the Weierstrass approximation theorem ). Now we have only to show that each p n ∗ ( x ) {\displaystyle p_{n}^{*}(x)} may be obtained by means of interpolation on certain nodes.
For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers ...
Here is an example, in C-like pseudocode, of an integer variant computed from some upper bound on the number of iterations remaining in a while loop. However, C allows side effects in the evaluation of expressions, which is unacceptable from the point of view of formally verifying a computer program.