Metric Definition of Sequence Convergence

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Hello! Last time, you used balls to decide whether a point belongs to the closure of a set: every ball around the point had to meet the set. Now we ask a more structured question. Given a particular sequence of points, do its terms eventually stay as close as we wish to a proposed limit?

This lesson develops the metric definition of convergence and uses it to settle examples in $R$ , Euclidean space, a discrete space, and a space whose points are functions. We will also see how to show directly that a sequence does not converge—and why the space containing the proposed limit matters.

What “eventually close” says

Let $(X, d)$ be a metric space and let $(x_{n})$ be a sequence in $X$ . We say that $x_{n}$ converges to $x \in X$ if

for every ε > 0, there is an N \in N such that n \geq N ⟹ d (x_{n}, x) < ε .

We write $x_{n} \to x$ . Equivalently, for every ball $B_{X} (x, ε)$ , all terms from some index onward lie inside that ball.

The order of the quantifiers does the work. You may choose a smaller radius, requiring a later cutoff $N$ . But once that cutoff is chosen, every term after it must pass the distance test. Earlier terms are unrestricted; a sequence need not move steadily toward its limit.

The blue points depict terms gathering around $x$ in the metric space $M$. For convergence, every neighborhood of $x$ must contain all sufficiently late terms. The $N$ printed inside the shaded region labels a neighborhood, not the cutoff index in the definition.

Read Definition 5 and its explanation in Notes on Metric Spaces. In particular, follow the shift from an inequality about distances to a statement about eventual membership in any ball.

Notes on Metric Spaces

Read the convergence definition and its ball interpretation to connect this lesson’s quantifiers with the balls used previously.

Find Definition 5, around pp. 4–5. Read the definition and explanation. Stop before Example 10; focus on why the cutoff may depend on the radius.

For a visual rehearsal on the real line, watch Michael Penn’s Real Analysis | Sequences and the ε-N definition of convergence. Its proof method applies unchanged when absolute value is replaced by another metric.

Real Analysis | Sequences and the ε-N definition of convergence.

Michael Penn draws the “eventually inside” picture and works through how to choose a cutoff index.

Watch definition and strategy for the quantifiers and the distinction between exploratory algebra and a finished proof. Then watch a worked proof for the sequence 1/n^2. Notice that the proof chooses N after \varepsilon is given and checks every n\ge N.

Turning a candidate limit into a proof

In the usual metric on $R$ , consider

x_{n} = 3 + \frac{2}{n} .

The candidate limit is $3$ . To test it directly, calculate the distance first:

d (x_{n}, 3) = 3 + \frac{2}{n} - 3 = \frac{2}{n} .

Given $ε > 0$ , the Archimedean property lets us choose an integer $N > 2/ ε$ . For every $n \geq N$ ,

d (x_{n}, 3) = \frac{2}{n} \leq \frac{2}{N} < ε .

Thus $x_{n} \to 3$ . The exploratory step was identifying how large $n$ must be; the proof then verified that a single choice of $N$ controls the entire tail.

Not every useful distance calculation is monotone. Let

y_{n} = \frac{( - 1 ) ^{n}}{n} .

The signs alternate, but

d (y_{n}, 0) = ∣ y_{n} ∣ = \frac{1}{n} .

Choosing $N > 1/ ε$ proves $y_{n} \to 0$ . Alternation alone does not prevent convergence: what matters is whether the distance to the candidate limit becomes and remains small.

The same approach works for vectors. In Euclidean $R^{2}$ , set

z_{n} = (\frac{1}{n}, \frac{( - 1 ) ^{n}}{n}) .

Its distance from $(0, 0)$ is

d (z_{n}, (0, 0)) = \frac{1}{n ^{2}} + \frac{1}{n ^{2}} = \frac{2}{n} .

Given $ε > 0$ , choose $N > 2 / ε$ . The resulting inequality for every $n \geq N$ proves $z_{n} \to (0, 0)$ . Even in a vector space, the proof is fundamentally a metric calculation.

There is a useful link to the previous lesson. The point $0$ lies in the closure of $A = (0, 1) \subseteq R$ because every ball around $0$ meets $A$ . The specific points $1/ (n + 1) \in A$ do more: they give an ordered sequence that converges to $0$ . Closure concerns what points can be found arbitrarily close; convergence checks whether one sequence eventually meets every closeness requirement.

When changing the metric changes the test

Recall that in the discrete metric, distinct points have distance $1$ , while identical points have distance $0$ . Suppose $x_{n} \to x$ . Apply the definition with $ε = 1$ . For some $N$ , every $n \geq N$ satisfies $d (x_{n}, x) < 1$ . The only possible distance is then $0$ , so $x_{n} = x$ for all $n \geq N$ . Conversely, any sequence that is eventually equal to $x$ converges to $x$ , since its distance from $x$ is eventually zero.

Hence, in a discrete metric space, convergent means eventually constant. Compare this with $1/ n \to 0$ in the usual metric on $R$ : its terms need never equal $0$ . The definition has not changed; the distances available in the space have.

A metric space can even have functions as its points. Let $C_{b} (R)$ be the bounded continuous real-valued functions, with sup metric

d_{\infty} (f, g) = t \in R sup ∣ f (t) - g (t) ∣.

For $f_{n} (t) = sin (t) / n$ and the zero function $0 (t) = 0$ ,

d_{\infty} (f_{n}, 0) = t \in R sup \frac{sin t}{n} = \frac{1}{n} .

So $f_{n} \to 0$ in this metric by the same cutoff argument as before. Here the condition $d_{\infty} (f_{n}, 0) < ε$ controls the error at every value of $t$ at once. Checking the metric tells you exactly what kind of closeness convergence requires.

Read Examples 12 and 13 in Notes on Metric Spaces alongside these two cases. They are worth reading as proofs: notice where each argument substitutes the relevant distance into the same definition.

Notes on Metric Spaces

Examples 12 and 13 contrast a metric in which closeness forces equality with one that measures the largest difference between functions.

Find Examples 12 and 13, around pp. 5–6. Starting where Example 12 says a discrete space is under discussion, read through the end of Example 13’s function calculation. In each example, identify the chosen \varepsilon, the resulting N, and what d(x_n,x)<\varepsilon means concretely.

Showing divergence—and checking the space

To prove that $x_{n}$ does not converge to a proposed point $x$ , negate the definition carefully:

there is an ε > 0 such that, for every N, some n \geq N satisfies d (x_{n}, x) \geq ε .

The radius may depend on $x$ . To prove that a sequence diverges in $X$ , this failure must be established for every candidate $x \in X$ .

Consider $x_{n} = (- 1)^{n}$ in $R$ . Fix any proposed limit $x$ . If $x \leq 0$ , choose an even $n \geq N$ ; then $x_{n} = 1$ and $∣ x_{n} - x ∣ \geq 1$ . If $x > 0$ , choose an odd $n \geq N$ ; then $x_{n} = - 1$ and $∣ x_{n} - x ∣ > 1$ . Either kind of index can be found beyond every $N$ . Thus $ε = 1$ defeats every candidate, and the sequence diverges. This differs from $y_{n} = (- 1)^{n} / n$ : there the oscillation remains, but its distance from zero shrinks.

Finally, specify the space in which a limit is allowed to live. The sequence

u_{n} = 2 - \frac{1}{n}

lies in $Y = (0, 2)$ for every $n \geq 1$ and converges to $2$ when regarded as a sequence in $R$ , since $∣ u_{n} - 2∣ = 1/ n$ . But it does not converge in $Y$ : $2 \in / Y$ . Could it have a different limit $x \in Y$ ? No. The induced metric gives the same distances between these points as the real metric, so convergence to $x$ in $Y$ would also be convergence to $x$ in $R$ . Metric limits are unique.

Here is the short proof of that last fact. If $x_{n} \to x$ and $x_{n} \to y$ , then for any $ε > 0$ , choose an index late enough that both $d (x_{n}, x) < ε /2$ and $d (x_{n}, y) < ε /2$ . The triangle inequality gives

d (x, y) \leq d (x, x_{n}) + d (x_{n}, y) < ε .

Since this holds for every positive $ε$ , $d (x, y) = 0$ , hence $x = y$ . The proof is a useful model for reading analysis: identify what must become zero, then use the metric and the quantifiers to make it arbitrarily small.

For another carefully negated divergence proof and the induced-space example, read the corresponding passage in Notes on Metric Spaces.

Notes on Metric Spaces

Example 14 shows how to rule out every proposed limit; the paragraph following it shows why a limit must belong to the space under discussion.

Find Example 14 and the following paragraph beginning “We emphasize,” around pp. 8–9. Read the divergence and subspace examples, stopping before Proposition 3. Track the order “choose a candidate limit, then choose a radius, then defeat every cutoff.”

Takeaways

To determine convergence directly, first name the metric space and a candidate limit. Compute or bound $d (x_{n}, x)$ ; then, for an arbitrary $ε > 0$ , choose $N$ and verify the bound for all $n \geq N$ . To disprove convergence, find a positive radius for which terms outside the ball keep appearing arbitrarily late, and rule out every candidate limit in the space.

The examples show why it pays to start with the metric rather than a visual guess: small oscillations can converge, discrete convergence forces eventual equality, and the sup metric requires simultaneous control across a function’s domain. Next, we will use convergence to investigate Cauchy sequences and completeness—including sequences whose terms grow close to one another but whose apparent limit is missing from the space.