Bounded and monotone
A bounded sequence is one whose terms never escape a fixed window: there is a number M with |a_n| <= M for all n. A first basic fact: every convergent sequence is bounded. (Past some N the terms sit within 1 of L, so they are bounded by |L| + 1; the finitely many earlier terms have a largest absolute value; take the max.) The converse fails — (-1)^n is bounded but diverges — so boundedness alone is not enough.
A monotone sequence is one that only ever goes one way: increasing (a_{n+1} >= a_n for all n) or decreasing (a_{n+1} <= a_n for all n). Monotone sequences cannot oscillate, so the only ways they can fail to converge is by running off to infinity. That observation is the heart of the next theorem.
The monotone convergence theorem
The monotone convergence theorem says: a sequence that is increasing and bounded above converges, and its limit is the supremum of its terms. (Symmetrically, decreasing and bounded below converges to the infimum.) This is the first theorem that conjures a limit without anyone telling you the answer in advance — and it works precisely because the reals are complete, via the least upper bound property.
Theorem: (a_n) increasing and bounded above => converges to L = sup{a_n}.
Proof:
The set S = {a_n : n in N} is nonempty and bounded above,
so by the least upper bound property L = sup S exists.
Let e > 0. Since L - e is NOT an upper bound (L is the LEAST one),
some term a_N satisfies a_N > L - e.
For all n > N, monotonicity gives a_n >= a_N > L - e.
Also a_n <= L for every n (L is an upper bound), so a_n <= L < L + e.
Combining: L - e < a_n <= L, hence |a_n - L| < e.
Therefore a_n -> L. QEDThe algebra of limits and the squeeze
Once you know a few limits, the algebra of limits lets you combine them without returning to epsilon. If a_n -> A and b_n -> B then: a_n + b_n -> A + B; a_n - b_n -> A - B; a_n b_n -> AB; and a_n / b_n -> A/B provided B is not 0 (and the terms are eventually nonzero). Each is proved once, with epsilon, then reused forever — that is the payoff of building the definition carefully.
The squeeze theorem is the other workhorse: if a_n <= b_n <= c_n eventually, and a_n -> L and c_n -> L, then b_n -> L too. It evaluates limits you cannot attack directly by trapping the unknown sequence between two known ones.
Evaluate d_n = (3n^2 + 2n) / (n^2 + 5) using the algebra of limits. Divide top and bottom by n^2: d_n = (3 + 2/n) / (1 + 5/n^2). Known facts: 2/n -> 0 and 5/n^2 -> 0 (from 1/n -> 0 and products). Numerator -> 3 + 0 = 3. Denominator -> 1 + 0 = 1 (nonzero!). By the quotient rule: d_n -> 3/1 = 3. Squeeze example: e_n = (sin n) / n. Since -1 <= sin n <= 1, we have -1/n <= e_n <= 1/n. Both -1/n -> 0 and 1/n -> 0, so by the squeeze e_n -> 0.