When does a sequence converge?

Forget the fancy word. A sequence is just an infinite list — a first number, a second, a third, and so on with no end:

a₁,  a₂,  a₃,  a₄,  a₅,  …

Each aₙ is a real number, and n is just its position in the queue: 1st, 2nd, 3rd. Formally a sequence is nothing more than a function from the counting numbers to the reals, a : ℕ → ℝ — feed it a slot number, get back the term sitting there.

If you've ever built an infinite Stream you already have the exact mental model. A sequence is a LazyList[Double]: values defined out to infinity, computed only when you ask for the next one.

Here's the whole idea in one sentence: a sequence converges when its terms eventually stop wandering and huddle around one number. That number is the limit.

Take 1, ½, ⅓, ¼, ⅕, … — the sequence aₙ = 1/n. The terms shrink, and past a while they're all packed right up against 0. They never actually hit 0, but that doesn't matter. What matters is that they get — and stay — arbitrarily close. We say the sequence converges to 0, written lim aₙ = 0 or aₙ → 0.

The two words doing all the work are eventually and stay. It's not enough for a term or two to land near the limit — every term from some point onward has to stay near it, and stay nearer the further out you go. A sequence that dips close and then bounces away hasn't converged.

"Eventually gets and stays close" is the right intuition, but a skeptic can poke at it: how close? How far out? The 19th-century fix answers both with a little challenge-and-response game between you and a doubter.

The doubter names a tolerance — a tiny positive distance, called ε ("epsilon"). Think of it as a target radius around the limit. Your job: prove that the sequence eventually fits inside that radius and never leaves. You do it by producing a cutoff N — a slot number — and promising that every term past slot N sits within ε of the limit.

aₙ → L means: for every ε > 0 there is an N such that
for all n > N,  |aₙ − L| < ε.

Read |aₙ − L| < ε as "the distance from term aₙ to the limit L is under ε." The definition says: no matter how mean the doubter gets with ε, you can always answer with an N. Smaller ε usually needs a bigger N — you have to walk further out before the terms are that tightly packed — but an N always exists. That "always" is convergence.

The definition is much friendlier as a picture. Plot the sequence: slot number n across the bottom, term value aₙ up the side, so every term is a dot. Draw a horizontal line at the limit L, then shade a band of half-width ε above and below it. That band is the doubter's tolerance made visible — the strip the terms are supposed to fall into.

Now the whole ε–N game is one sentence you can see: can you slide a vertical cutoff line to a spot where every dot to its right lands inside the band? If yes, that cutoff is your N. If the band is thin, you'll have to push N further right — but for a convergent sequence you can always find one. For a divergent one, some dots keep poking out of the band no matter where you put the line.

Drag the two handles below: the ε knob on the right shrinks or grows the band, and the N handle on top slides the cutoff. Try each sequence.

Notice the pattern: for 1/n or (1+1/n)ⁿ, shrinking ε just means dragging N a little further right — the tail always fits eventually. That's a limit. The last sequence never lets you win, no matter how you drag. That's our next topic.

A sequence that doesn't converge is called divergent. There's no drama to the definition — it's just "not convergent" — but it shows up in two flavours worth telling apart.

It oscillates. Take aₙ = (−1)ⁿ = −1, 1, −1, 1, …. It never wanders off to infinity — it stays bounded between −1 and 1 — but it never settles either. Try to trap it: pick any candidate limit and any band thinner than the gap between −1 and 1, and half the terms will always sit outside it, forever, past every cutoff. No single number can be the limit, so there is none.

It runs off. Take aₙ = n = 1, 2, 3, 4, …. The terms march off toward infinity and never come back. We write aₙ → ∞ as shorthand, but be clear: ∞ is not a number the sequence is converging to — it's just a description of the terms growing past every bound. This sequence diverges too.

The lesson from both: bounded is not enough. Staying in a finite range keeps you from running off, but you can still rattle around forever without homing in. Convergence needs more than a ceiling — which is exactly the gap section 7 fills.

Good news for anyone who doesn't want to run the ε–N game every single time: limits respect arithmetic. If you already know two sequences converge, you get their combinations for free. Suppose aₙ → A and bₙ → B. Then:

• Sums: aₙ + bₙ → A + B. The limit of a sum is the sum of the limits.
• Products: aₙ · bₙ → A · B — and scaling, c·aₙ → c·A, is just the constant case.
• Quotients: aₙ / bₙ → A / B, as long as B ≠ 0 (you can't divide by a limit of zero and expect sense).

These are the workhorses. To find lim (2 + 1/n) you don't touch ε at all: 2 → 2, 1/n → 0, so by the sum law the whole thing goes to 2. Composability, exactly like folding pure functions — once you trust the pieces, the combinations come for free.

Here's a test you can often check at a glance, no ε in sight. Suppose a sequence only ever climbs (or stays level) — it's increasing, or monotone — and it's also bounded above by some ceiling it never crosses. Then it must converge. A sequence that only goes up but can't go up forever has nowhere left to do but home in on a limit.

And here's the punchline that ties this chapter to the last one: the limit it lands on is exactly the least upper bound. The terms climb toward their tightest ceiling — their sup — and that sup is the limit. Same story for a decreasing-and-bounded-below sequence: it converges down to its greatest lower bound.

But wait — a bounded, climbing sequence of rationals can have its sup fall into a hole (remember 1, 1.4, 1.41, … climbing toward √2). The theorem only works because we're in ℝ, where every bounded set is guaranteed a real sup to converge to. This test is completeness, wearing a different hat.

The ε–N definition has one catch: to use it, you have to already know the limit L. You're measuring |aₙ − L|. But what if you suspect a sequence settles down and have no idea what number it's settling on? You need a convergence test that never mentions the destination.

That's the Cauchy idea, and it's a small but brilliant shift: instead of asking whether the terms get close to the limit, ask whether the terms get close to each other. A sequence is Cauchy when, for every tolerance ε, there's a cutoff N past which any two terms are within ε of one another — |aₘ − aₙ| < ε for all m, n > N. The tail collapses in on itself.

You met this exact notion last chapter under the name "huddle." The magic — and it's completeness once more — is that in ℝ the two notions are the same: a sequence of reals converges if and only if it is Cauchy. So you can certify convergence entirely from the inside, by watching the terms bunch up, without ever naming where they land.

A programmer approximates a limit the Cauchy way, because you can actually run it: iterate the sequence and stop when two successive terms sit within eps of each other. You never need to know the answer up front — you watch the terms stop moving. Below, we pin down e as the limit of (1 + 1/n)ⁿ. Flip between the two languages; the shape is identical.