ε-N Convergence

Making "gets close to" precise: for every tolerance, a threshold that works.

The idea

The $\varepsilon$–$N$ definition states precisely what it means for a sequence to converge; it replaces the loose phrase "gets close to $L

Definition (Convergence of a sequence).

A sequence $a_{n}$ converges to the limit $L$ if $\forall \varepsilon > 0 \ \exists N \ \forall n \ge N : \ |a_{n} - L| < \varepsilon,$ read aloud as: for every $\varepsilon$ greater than zero there exists an $N$ such that for all $n$ at least $N$, the distance from $a_{n}$ to $L$ is less than $\varepsilon$.

Whatever tolerance $\varepsilon$ is demanded, some threshold $N$ answers it: from the $N$-th term onward, every term stays within $\varepsilon$ of $L$ — in the figure, every term to the right of $N$ lies inside the band between $L - \varepsilon$ and $L + \varepsilon$, while several earlier terms fall outside it.

The order of the three quantifiers carries the meaning. Because $\exists N$ stands to the right of $\forall \varepsilon$, we choose the threshold after the tolerance is known — and normally must, since a tighter tolerance forces a later threshold. Because $\exists N$ stands to the left of $\forall n$, the one chosen $N$ must cover every term beyond it; a single stray term past $N$ defeats the choice.

To meet a named tolerance, write $|a_{n} - L|$ as an expression in $n$ and solve the inequality $|a_{n} - L| < \varepsilon$ for $n$.

Ways to work on it

• Walkthrough. The definition, then finding a threshold for a given tolerance.
• Practice. Given a sequence and tolerance, find the smallest threshold.
• Hardest. Find a threshold for a sequence with a messier error term.

Not sure where to start? Take the ten-question placement test.