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Just as a convergent sequence in **R** can be thought of as a sequence of better and better approximtions to a limit, so a sequence of "points" in a metric space can approximate a limit here.

**Definition**- A sequence (
*x*_{n}) of points in a metric space (*X*,*d*) converges to a limit*α*if the*real sequence*(|*d*(*x*_{n},*α*)| converges to 0 in**R**. **Remark**- If you insist on "back to basics" this reads:

Given*ε*> 0 there exists*N*∈**N**such that if*n*>*N*then we have*d*(*x*_{n},*α*) <*ε*. **Examples**- For
**R**with its usual metric this is the same as before. - In
**C**with the metric*d*(*z*,*w*) = |*z*-*w*|, consider the sequence (*z*,*z*^{2},*z*^{3}, ...) with |*z*| < 1.

Then this sequence converges to 0 ∈**C**.e.g. Take

*z*=^{(1+i)}/_{2}so that |*z*| =^{1}/_{√2}

The points lie on a spiral.

**Proof that the sequence converges**Look at the real sequence (

*d*(*x*_{n}, 0)) = (|*z*^{n}- 0|) = (|*z*|^{n})→ 0 since |*z*| < 1.

- Look at the sequence in
**R**^{2}given in the following way.

*x*_{1}= 2√3,*y*_{1}= 3 and then define the later terms by 2/*x*_{n+1}= 1/*x*_{n}+ 1/*y*_{n}and*y*_{n+1}= √(*x*_{n+1}*y*_{n}).

This gives a sequence which evaluates numerically to:

( (3.4642, 3.0000), (3.2154, 3.1057), (3.1596, 3.1325), (3.1460, 3.1392), (3.1426, 3.1408), 3.1418, 3.1414), ...)This sequence is based on the method used by Archimedes to calculate π.

Start with*x*_{1}= the semiperimeter of the "outside hexagon"

Start with*y*_{1}= the semiperimeter of the "inside hexagon"

and then double the number of sides to get a 12-gon , a 24-gon, etc.

Archimedes took the calculation up to*n*= 5 (corresponding to a 96-gon).In fact the sequence in

**R**^{2}converges to the point (π, π).

- For
- This last result suggests the following.
**Theorem**

*Convergence in***R**^{2}*with its usual metric d*_{2}*is "componentwise"*.That is ((

*x*_{1},*y*_{1}), (*x*_{2},*y*_{2}), ... )→ (*α*,*β*) if and only if

(*x*_{1},*x*_{2}, ... )→*α*and (*y*_{1},*y*_{2}, ... )→*β*.**Proof**

- ( ⇒ ) Given
*ε*> 0 we know that we have*N*so that if*n*>*N*then √(|*x*_{n}-*α*|^{2}+ |*y*_{n}-*β*|^{2}) <*ε*. But then we must have |*x*_{n}-*α*| <*ε*and so (*x*_{n})→*α*. Similarly for the other component.Conversely: Given

*ε*> 0 choose*N*so that if*n*>*N*then |*x*_{n}-*α*| <*ε*and |*y*_{n}-*β*| <*ε*. But then √(|*x*_{n}-*α*|^{2}+ |*y*_{n}-*β*|^{2}) < √(*ε*^{2}+*ε*^{2}) = √2*ε*and so we may make this as small as we like.

- In fact this last result holds for any
*finite-dimensional*space**R**^{n}and also holds for such spaces with*any*of the metrics*d*_{p}. The situation for*infinite-dimensional*spaces of sequences or functions is different as we will see in the next section.

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