Compute the consecutive difference aₙ₊₁ − aₙ. This difference is nonnegative wherever the sequence does not decrease, and it is positive wherever the increase is strict. A zero difference identifies a repeated adjacent value rather than a failure of monotonicity. Checking the sign for every relevant index provides a direct algebraic test.
For an increasing sequence, an upper bound prevents the terms from rising indefinitely. The sequence then approaches a finite limit, so boundedness connects local comparisons of neighboring terms with long-term behavior. If no upper bound exists, the terms grow without limit instead. This distinction is central when analyzing convergence.
Equality between consecutive terms is compatible with an increasing sequence but not with a strictly increasing one. A sequence may therefore remain temporarily flat while preserving its overall monotone behavior. This distinction matters when interpreting recursive updates or inequalities, because a zero change can be acceptable even when positive growth is not required.
In a recursive sequence, examine the rule that produces aₙ₊₁ from aₙ. Rewriting the update as aₙ₊₁ = aₙ plus an increment exposes whether each step adds a nonnegative or positive amount. This connects the recursion to consecutive differences and can establish monotonic behavior symbolically for every relevant index.
To test a proposed sequence, compare adjacent expressions, form aₙ₊₁ − aₙ, and determine its sign for every relevant index. Nonnegative values support an increasing conclusion, while strictly positive values support a strict conclusion. Afterward, search for an upper bound, since the monotonicity test alone does not determine whether a finite limit exists.
Monotone sequences can organize approximations by showing whether successive terms move upward without reversal. In recursive models of cumulative change, the same comparison tracks whether each new value adds or preserves progress. Pairing that behavior with an upper-bound check helps distinguish a process approaching a finite value from one that grows without limit.