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We will consider some more examples of convergence in metrics.

First, we look at some examples of convergence in spaces of sequences.

- Let
*X*be the set of all bounded real sequences with the metric*d*_{∞}defined earlier.

Take the sequence of "points" in*X*(that is, a sequence of sequences) given by:= (*x*_{1}^{1}/_{1},^{1}/_{2},^{1}/_{3}, ... ),= (*x*_{2}^{1}/_{12},^{1}/_{22},^{1}/_{32}, ... ),= (*x*_{3}^{1}/_{13},^{1}/_{23},^{1}/_{33}, ... ) , ...

Then this sequence of sequences converges to the sequence= (1 , 0 , 0 , 0 , ... ) in*y**d*_{∞}.**Proof**

*d*_{∞}(,*x*_{n}) = lub { 0 ,*y*^{1}/_{2n},^{1}/_{3n}, ... } =^{1}/_{2n}and this is small when*n*is large.

- This next example shows one of the differences between the finite and infinite-dimensional cases.
Take the sequence in

*X*given by:= (1 , 0 , 0 , ... ),*x*_{1}= (0 , 1 , 0 , ... ),*x*_{2}= (0 , 0 , 1 , 0 , ... ), ...*x*_{3}

Then although the "sequence of first components", the "sequence of second components", the "sequence of third components", ... all converge to 0, the sequencedoes*x*_{n}*not*converge to the zero sequence:**0**, since*d*_{∞}(,*x*_{n}**0**) = 1 for all*n*.So the theorem proved for finite dimensional spaces in the last section does not hold for this infinite dimensional space.

- Define a sequence of functions in
*C*[0, 1] by*f*_{1}(*t*) =*t*,*f*_{2}(*t*) =*t*/_{2},*f*_{3}(*t*) =*t*/_{3}, ...

We claim that this sequence converges to the 0-function in the metric*d*_{∞}.**Proof**

The maximum of the function |*f*_{n}(*x*) - 0| is at*x*= 1 and is^{1}/_{n}. Thus the real sequence (*d*_{∞}(*f*_{n}, 0))→ 0 and so the 0-function is the limit.

**Remarks**

We can get a "picture" of what convergence in the metric*d*_{∞}"Looks like".

Given a function*g*on (say) [0, 1], we can draw "An*ε*-band" around its graph.Then a function is within

*ε*of*g*provided its graph lies in this*ε*band.Thus a sequence (

*f*_{n})→*g*in*d*_{∞}if the graphs of every function "far enough down" the sequence lie in an arbitrarily small*ε*band.

- This time we consider converence in the
*d*_{1}metric given by*d*_{1}(*f*,*g*) = |*f*(*x*) -*g*(*x*)|*dx*. (i.e. the "distance between the functions is the*area*between their graphs").- The picture above shows that the sequence of functions (
*x*/_{n}) converges to the 0-function.

- Here is a curious sequence of "spike functions"

Define*f*_{n}(*x*) by:Then (

*f*_{n})→ the 0-function in the metric*d*_{1}.

**Proof**

*d*_{1}(*f*_{n}, 0) = the area under the curve =^{1}/_{n}and so (*d*_{1}(*f*_{n}, 0))→ 0 and the sequence converges.

Note however, that this sequence does not converge to the 0-function in*d*_{∞}since the graph of*f*_{n}"sticks out" of any small*ε*band of the 0-function.Thus it can happen that a sequence which converges in one metric may fail to converge in another.

- The picture above shows that the sequence of functions (

- Convergence of sequences of functions in the metric
*d*_{∞}is particularly important. It is called**uniform convergence**and it turns out that this convergence has particularly nice properties, particularly in relation to differentiation and integration. - Convergence of sequences of functions in the metric
*d*_{1}is also important. It is called**convergence in the mean**(because in some sense it "Averages out" the distance between functions). It is important in some applications. For example, the partial sums of the Fourier series of a function converges to the function*in the mean*. - The most obvious way of considering convergence of a sequence of functions is
**pointwise convergence**. That is, a sequence (*f*_{n}) of functions in (say)*C*[0, 1] is*pointwise convergent*to*f*if at each*x*∈ [0. 1], the*real*sequence (*f*_{n}(*x*)) converges in**R**to*f*(*x*).

For example, the pointwise limit of the sequence of "spike functions" above is the function defined by*f*(^{1}/_{2}) = 1 and*f*(*x*) = 0 otherwise.

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