Convergence in a Metric Space¶

Definition¶

Given a metric space $(X, d)$, a sequence $(x_{i})_{i=1}^\infty$ and a point $x$ in $X$, the sequence is said to converge to $x$ under metric $d$ if $\displaystyle \lim_{i\to\infty}d(x_i, x) = 0$$0$.

Interpretation¶

A sequence converges to $x$ when its distance to $x$ tends to $0$. This is just like convergent sequences of real numbers, but with a general notion of distance: the definition reduces convergence in any metric space to the convergence of the real numbers $d(x_i,x)$ to $0$.

Written out, the definition says that for every $\varepsilon>0$ there is an $N$ with $d(x_i,x)<\varepsilon$ for all $i\ge N$$N$. In the figure the terms spiral in towards $x$; the shaded disc of points within distance $\varepsilon$ of $x$ misses some early terms such as $x_1$ and $x_2$, but holds every term from some index $N$ on. A smaller $\varepsilon$ may require a larger $N$.

A limit, when it exists, is unique: if $x_i\to x$ and $x_i\to y$$y$, the triangle inequality gives $d(x,y)\le d(x,x_i)+d(x_i,y)\to0$$0$, so $x=y$$y$. Whether a sequence converges depends on the metric. Under the discrete metric, $d_{\mathrm{disc}}(x_i,x)<1$ forces $x_i=x$$x$, so only sequences that are eventually constant converge; $x_i=1/i$ converges to $0$ for $|x-y|$ but not for $d_{\mathrm{disc}}$. Strongly equivalent metrics, such as the Euclidean and taxicab metrics in the Examples, have the same convergent sequences.

A common slip is to confuse converging with terms getting close to each other. The latter is the Cauchy property, and in a space with missing points, such as $\mathbb{Q}$, a Cauchy sequence need not converge.

Key Results¶

• A convergent sequence in a metric space must be a Cauchy sequence.

Examples¶

• The sequence $(\boldsymbol{x}_{i})_{i=1}^\infty$ where $\boldsymbol{x}_i = (1/i, 1/i)$ in $\mathbb{R}^2$ converges to $(0,0)$ under the Euclidean metric $d_2$.
• The sequence $(\boldsymbol{x}_{i})_{i=1}^\infty$ where $\boldsymbol{x}_i = (1/i, 1/i)$ in $\mathbb{R}^2$ also converges to $(0,0)$ under the taxicab metric $d_1$.

Prerequisites¶

• Metric and Metric Space
• Limit of a Sequence

Related Concepts¶

• Strongly Equivalent Metrics
• Discrete Metric
• Cauchy Sequence in a Metric Space
• Accumulation Point in a Metric Space

Sources¶

• Math III: Analysis · Lecture 4

Symbols: epsilon, infinity, rational numbers, real numbers

Last updated: 20 October 2026 03:02