How can distance define a limit?

Source scope: §3.3. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Two routes between the same points

From $(0,0)$ to $(3,4)$, a straight route has length $5$. A horizontal-then-vertical route has length $7$. These are Euclidean and city-block metrics; neither changes the endpoints. A metric need not come from a vector space.

A metric on a set $S$ is a function $d:S\times S\to\mathbb R$ such that, for all $x,y,z\in S$,

1. $d(x,y)\ge0$, with equality exactly when $x=y$.
2. $d(x,y)=d(y,x)$.
3. $d(x,z)\le d(x,y)+d(y,z)$, the triangle inequality.

A tempting substitute, squared Euclidean distance, fails the third condition: on the real line, $d(0,2)=4$ exceeds $d(0,1)+d(1,2)=2$.

A tail that stays close

A sequence $(x_n)$ in $S$ converges to $x$ if

The entire tail must stay inside the distance threshold. One close point is insufficient. Real-number convergence is the special case $d(x,y)=|x-y|$.

For $x_n=1/n$ and $x=0$, choose any integer $N>1/\epsilon$. Then $n\ge N$ implies $1/n\le1/N<\epsilon$. For $\epsilon=0.1$, $N=11$ works; $N=10$ fails the strict inequality at its first term.

Transfer

Does the sequence $0,1,0,1,\ldots$ converge to zero because it reaches zero infinitely often?