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A real increasing sequence bounded above converges to its supremum; a decreasing sequence bounded below converges to its infimum. For the increasing case, the definition of supremum provides a term above $L-\epsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic monotone convergence theorem for integrals.

A fractional Brownian motion with Hurst exponent $H\in (0,1)$ is a centered Gaussian process with the displayed standard covariance function and a continuous modification. Its increment variance is $|s-t{|}^{2H}$. Some sources omit the factor $1/2$, multiplying the process by $\sqrt{2}$; the corresponding increment variance is then $2|s-t{|}^{2H}$. Its canonical pseudometric of a Gaussian process is proportional to $|s-t{|}^H$, which gives finite expected value of its absolute supremum on compact intervals through the Dudley entropy integral.

The independent constraints $|{\phi }_0|\le 1$ and $|\phi |\le 1$ separate the dual supremum into $|\Omega |+|Du|(\Omega )$. They do not define square-root graph area. For $|\nabla u|=1$ the box-constrained value is $2|\Omega |$, while the relaxed graph-area functional is $\sqrt{2}|\Omega |$.

The least-upper-bound property states that every nonempty set of real numbers that has an upper bound has a supremum in the real numbers. It is an order-theoretic formulation of the completeness of the real numbers.

For a continuous path $\gamma :[a,b]\to X$ in a metric space, take the supremum over finite partitions $a=t_0<\cdots <t_n=b$ of ${\sum }_{i=1}^{n}d(\gamma (t_i),\gamma (t_{i-1}))$. This definition allows $+\infty$ and needs no derivative. A metric geodesic parametrized on $[a,b]$ has length $b-a$. In a Riemannian manifold, it agrees with the usual arc length integral for a smooth curve.

Every nonempty bounded-above subset of a total order has a least upper bound. In the real line this is the supremum property used to extend maps defined on an order-dense subset.