Convergent and divergent sequences: fundamental properties

Uniqueness of the Limit

If a sequence converges, its limit is unique. Proof by contradiction: if u_n → l and u_n → l' with l ≠ l', let ε = |l – l'|/2 > 0. The intervals [l − ε, l + ε] and [l − ε, l’ + ε] are disjoint, whereas u_n should belong to both from a certain order onwards: contradiction.

Convergence implies boundedness

Every convergent sequence is bounded. Note that the converse is false! The sequence $u_n = (-1)^n$ is bounded (between -1 and 1) but does not converge: it oscillates without ever stabilising.

Operations on limits

If u_n → l and v_n → l' (finite limits), then:

Operation	Limit
u_n + v_n	l + l'
u_n * v_n	l * l'
u_n / v_n (if l' ≠ 0)	l / l'
λ * u_n	λ * l

These rules assume that l and l' are finite. As soon as a limit is infinite, caution is required: indeterminate forms (infinity – infinity, 0 * infinity, infinity/infinity, 0/0) cannot be evaluated term by term and require specific analysis (factorisation, conjugate quantity, limit expansion, etc.).

Divergence

A sequence diverges if it does not have a finite limit. We distinguish between:

• divergence towards +∞ or -∞ (the sequence ‘explodes’);
• divergence due to oscillation (such as (-1)^n), without even an infinite limit.

A classic pitfall

Do not confuse ‘diverging towards +∞’ with ‘diverging’ in the general sense. A sequence that tends towards +∞ is indeed divergent in the strict sense (no finite limit), but its asymptotic behaviour is perfectly determined, unlike that of an oscillating sequence.