Sequences & Limits | Real Analysis

The Language of Limits

Sequences are the foundation of analysis. A sequence converges to a limit L if its terms eventually get arbitrarily close to L and stay there. The precise epsilon-N definition makes this intuition rigorous:

For every ε > 0, there exists N such that n > N implies |aₙ - L| < ε

This section will help you visualize what this definition really means and build intuition for working with limits.

The ε-N Definition of Convergence

The heart of rigorous analysis: for a sequence to converge to L, we must be able to trap all sufficiently late terms within any tolerance ε of L. Drag the epsilon slider to see how the required N changes—smaller tolerance demands going further into the sequence.

Select Sequence:

Show annotations

Epsilon (ε): 0.300

0.01 (tight)2.0 (loose)

Current sequence: aₙ = 1/n

Limit: L = 0.0000

Within ε of limit

Outside ε band

N marker

ε-neighborhood

Understanding the ε-N Definition

The green band shows all values within ε of the limit L
The red dashed line marks N: all terms after this point must stay in the band
Try decreasing ε and watch how N must increase to compensate
A smaller ε means a stricter requirement, so we need to go further into the sequence

Convergent Sequence Explorer

Watch sequences approach their limits in real-time. Observe how different sequences converge at different rates—some approach quickly, others more gradually. The animation reveals the 'settling down' behavior that defines convergence.

Speed:

Term 0 / 60

Select Sequence:

1/n

The harmonic sequence converges to 0

Limit: L = 0

Monotonicity: decreasing

Why Sequences Fail to Converge

Not all sequences converge. Explore the three main ways a sequence can diverge: growing without bound (unbounded), oscillating between values without settling, or behaving chaotically within bounds. Compare with convergent sequences to see the difference.

Select Divergent Sequence:

Number of terms: 40

Compare with convergent sequence (1/n → 0)

Unbounded (→ +∞)

Terms grow without bound as n increases. No matter how large a number you pick, the sequence eventually exceeds it.

This sequence: Diverges to +∞ (unbounded)

Why Divergence Matters

Unbounded sequences grow without limit; no ε-neighborhood can contain their tails
Oscillating sequences have multiple cluster points but no single limit
Chaotic sequences are bounded but don't settle toward any value
Being bounded is necessary but not sufficient for convergence

The Monotone Convergence Theorem

A powerful result: every bounded monotone sequence must converge. An increasing sequence bounded above is 'squeezed' toward its supremum; a decreasing sequence bounded below approaches its infimum. This theorem is fundamental for proving convergence without knowing the limit in advance.

Sequence Type:

Bound: 2.00

Show supremum region

Number of terms: 30

Monotone Convergence Theorem (MCT)

Every bounded monotone sequence converges to its supremum (if increasing) or infimum (if decreasing).

If (aₙ) is increasing and bounded above, then lim aₙ = sup{aₙ}

Current sequence: aₙ = 2 - 1/n converges to 2.00

Why Does This Work?

An increasing sequence bounded above must have a least upper bound (supremum)
The sequence approaches but never exceeds this supremum
By the completeness of real numbers, this supremum must be the limit
Similarly, a decreasing sequence bounded below converges to its infimum

The Bolzano-Weierstrass Theorem

Every bounded sequence has a convergent subsequence. This demo visualizes the bisection proof: repeatedly halving the interval and selecting terms from the more populated half extracts a subsequence that must converge. This theorem connects boundedness to convergence in a deep way.

Sequence Type:

Animation Speed: 800ms

Outside interval

Inside interval

Selected for subsequence

Bolzano-Weierstrass Theorem

Every bounded sequence has a convergent subsequence.

The Bisection Method:

Start with interval [-1, 1] containing all terms
Bisect: divide interval in half at midpoint
Choose the half with more (or equal) terms
Select one term from that half for our subsequence
Repeat — the intervals shrink, forcing convergence