MT2002 Analysis

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## Convergence in a metric space

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 (xn) of points in a metric space (X, d) converges to a limit α if the real sequence (|d(xn, α)| converges to 0 in R.

Remark

If you insist on "back to basics" this reads:
Given ε > 0 there exists NN such that if n > N then we have d(xn, α) < ε.

Examples

1. For R with its usual metric this is the same as before.

2. In C with the metric d(z, w) = |z - w|, consider the sequence (z, z2, z3, ...) 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(xn, 0)) = (|zn- 0|) = (|z|n)→ 0 since |z| < 1. 3. Look at the sequence in R2 given in the following way.
x1= 2√3, y1= 3 and then define the later terms by 2/xn+1= 1/xn+ 1/yn and yn+1= √(xn+1yn).
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.
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 R2 converges to the point (π, π).

This last result suggests the following.

Theorem

Convergence in R2 with its usual metric d2 is "componentwise".

That is ((x1 , y1), (x2 , y2), ... )→ (α , β) if and only if
(x1 , x2 , ... )→ α and (y1 , y2 , ... )→ β.

Proof
( ⇒ ) Given ε > 0 we know that we have N so that if n > N then √(|xn- α|2+ |yn- β|2) < ε. But then we must have |xn- α| < ε and so (xn)→ α. Similarly for the other component.

Conversely: Given ε > 0 choose N so that if n > N then |xn- α| < ε and |yn- β| < ε. But then √(|xn- α|2+ |yn- β|2) < √(ε2+ ε2) = √2 ε and so we may make this as small as we like. In fact this last result holds for any finite-dimensional space Rn and also holds for such spaces with any of the metrics dp. The situation for infinite-dimensional spaces of sequences or functions is different as we will see in the next section.

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JOC September 2001