Sequence convergence

The definition requires the condition for every $\varepsilon > 0$ to ensure that the sequence terms can be made arbitrarily close to the limit. A single band or a finite plotted sample is insufficient; the cutoff $N$ may depend on $\varepsilon$ and need not be unique.

Conditions: The tolerance $\varepsilon$ is arbitrary and positive.; The cutoff $N$ depends on $\varepsilon$.; The condition must hold for all $n > N$.

The epsilon-N definition uses a horizontal tolerance band from $a - \varepsilon$ to $a + \varepsilon$ and a vertical cutoff $N$. For a given tolerance $\varepsilon$, a valid integer cutoff $N$ ensures that every term with index $n > N$ lies within the band.

Conditions: The candidate limit is $a$.; The tolerance is $\varepsilon > 0$.; The cutoff $N$ is an integer.

The candidate limit in the sequence visualization is the horizontal level that both the monotone-looking and oscillating sequences approach. This level serves as the center of the tolerance band in the epsilon-N definition.

Conditions: The visualization shows two sequences approaching the same value.; The tolerance band is centered around this value.

Early terms do not decide the limit because the epsilon-N definition only restricts the behavior of terms after a certain cutoff $N$. Finitely many exceptions before the cutoff do not affect convergence.

Conditions: The sequence has a candidate limit $a$.; The cutoff $N$ is chosen based on $\varepsilon$.; The definition applies to all $n > N$.

Convergence does not require monotonicity because the formal definition only restricts the behavior of sufficiently late terms. A sequence can oscillate with diminishing amplitude and still approach a limit, as long as all terms beyond a certain index fall within any given tolerance band around that limit.

Conditions: The sequence approaches a candidate limit $a$.; The tolerance band is defined by $\varepsilon > 0$.; The cutoff $N$ depends on $\varepsilon$.

The mathematical formula for the epsilon-N definition of convergence is $\forall\varepsilon>0,\ \exists N\in\mathbb N,\ \forall n>N:\ |a_n-a|<\varepsilon$. This means that for every positive tolerance, there exists a natural number cutoff such that all sequence terms after that cutoff are within the tolerance of the limit $a$.

Conditions: The sequence is $a_n$.; The limit is $a$.; $\varepsilon > 0$ is the tolerance.; $N \in \mathbb{N}$ is the cutoff.

The tolerance band from $a - \varepsilon$ to $a + \varepsilon$ is equivalent to the absolute value inequality $|a_n - a| < \varepsilon$. Distance less than $\varepsilon$ from the limit $a$ means that the term $a_n$ lies strictly inside the open band around $a$.

Conditions: The limit is $a$.; The tolerance is $\varepsilon > 0$.; The sequence term is $a_n$.

The horizontal axis records the term index $n$, while the vertical axis records the value of the sequence term. This setup illustrates that a sequence gives discrete points rather than a continuous trajectory.

Conditions: The visualization is of a sequence.; The axes are labeled for index and value.