Comparison Theorems and Monotonic Convergence

The ‘sentries’ (or ‘enclosure’) theorem

If, from a certain order on, v_n ≤ u_n ≤ w_n, and if v_n → l and w_n → l (same limit), then u_n → l. This theorem is useful when u_n is difficult to analyse directly but can be bounded by two simple sequences with the same limit.

Example

Let u_n = sin(n)/n. Since -1 <= sin(n) <= 1, we have -1/n <= u_n <= 1/n. Now, -1/n -> 0 and 1/n -> 0, so by the ‘sentries’ theorem, u_n -> 0.

Bounded monotonic sequences

Fundamental theorem (a consequence of the completeness of R):

• every increasing sequence with an upper bound converges;
• every decreasing sequence with a lower bound converges.

Note: this theorem guarantees the existence of the limit, but does not give its value. Additional reasoning (fixed point, taking the limit in a recurrence relation) is often required to determine it precisely.

Adjacent sequences

Two sequences (u_n) and (v_n) are adjacent if u_n is increasing, v_n is decreasing, and (v_n - u_n) → 0. They then converge to the same limit. This is the key tool for bounding a real number (the dichotomy method, sequences approaching π, for example).

Cauchy’s criterion (mention)

A real sequence converges if and only if it is a Cauchy sequence: for any ε > 0, there exists an N such that for all p, q ≥ N, |u_p – u_q| ≤ ε. This criterion allows us to prove convergence without knowing the limit in advance.

Classic pitfall

Do not forget the upper bound condition: an increasing sequence that is NOT bounded from above diverges to +∞ (example: u_n = n^2).