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The definitions given earlier for **R** generalise very naturally.

**Definition**

The sequence (*x*_{n}) in a metric space is **convergent** to *x* *X* if:

One may rephrase this as

- The plane
**R**^{2}with various metrics:

It turns out that for the metrics d_{1}, d_{2}, d_{}if a sequence is convergent in one metric, it is convergent in the others.

In fact for these metrics a sequence ((*x*_{i},*y*_{i})) in**R**^{2}converges to (*x*,*y*) if and only if (*x*_{i})*x*and (*y*_{i})*y*in**R**. That is, convergence is*componentwise*.

However, you should note that for any set with the*discrete metric*a sequence is convergent if and only if it is eventually constant.

So whether the sequence converges or not*may*depend on what metric you are using.This becomes even more important in:

*C*[0, 1] with various metrics.

Take the sequence (*f*_{n}) with*f*_{n}the function whose graph is:

Then*d*_{1}(*f*_{1}, 0 ) =^{1}/_{n}(where 0 is the zero-function) and so this sequence converges to the zero-function in*d*_{1}.

However, in*d*_{}we have*d*_{}(*f*_{n}, 0) = 1 for all*n*and so this sequence does*not*converge to the zero-function in the metric*d*_{}. In fact it does not converge took any function.We will look at

*C*[0, 1] with the*d*_{}-metric in more detail later.

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