Why the sequence is not convergent?

**symbol0** · Jun20-09, 09:36 PM

In a book I am reading, they mention the following as an example of a Cauchy sequence which is not convergent:
Consider the set of all bounded continuous real functions defined on the closed unit interval, and let the metric of the set be
d(f,g)= $LaTeX Code: \\int_0^1 \\! |f(x)-g(x)| \\, dx$ .
Let

be a sequence in this space defined as:

if $LaTeX Code: 0 \\leq x \\leq 1/2$

if $LaTeX Code: 1/2 \\leq x \\leq 1/2+(1/2)^n$

if $LaTeX Code: 1/2+(1/2)^n\\leq x \\leq 1$

I can see that this is a Cauchy sequence, but I can't see how this sequence does not converge. I would say that it converges to:

if $LaTeX Code: 0 \\leq x \\leq 1/2$

if $LaTeX Code: 1/2 < x \\leq 1$

I would appreciate if anybody can help me to see why this sequence does not converge.

**Hurkyl** · Jun20-09, 10:08 PM

Originally Posted by symbol0

I can see that this is a Cauchy sequence, but I can't see how this sequence does not converge. I would say that it converges to:
if $LaTeX Code: 0 \\leq x \\leq 1/2$
if $LaTeX Code: 1/2 < x \\leq 1$

I'm going to assume you meant to say something like the following.

I would say that it converges to the function f defined by
if $LaTeX Code: 0 \\leq x \\leq 1/2$
if $LaTeX Code: 1/2 < x \\leq 1$ [/quote]

I would appreciate if anybody can help me to see why this sequence does not converge.

A good way to see these things is to try and prove that it does, and see where things fail. You need to:
(1) Show that f is well-defined
(2) Show that $LaTeX Code: \\lim_{n \\rightarrow +\\infty} d(f_n, f) = 0$
Try it out, and let us know how it goes!

**symbol0** · Jun21-09, 07:30 PM

Thanks for the reply Hurkyl,
First of all , I think this sequence is not really in the mentioned space because when n is even, the functions are not continuous.
But even if we take only the functions with odd n, I think the sequence converges to the function I wrote, but this function is not continuous, so it is not in the space.
Thus the sequence does not converge in the space.

Am I right?

**Hurkyl** · Jun21-09, 10:25 PM

Originally Posted by symbol0

I think the sequence converges to the function I wrote, but this function is not continuous, so it is not in the space.

That last part is the key thing here -- f is not in the space.

In fact, technically speaking, d was only defined for pairs of elements in the space, so it would be wrong to say that the sequence converges to f! You only get that convergence when looking at the larger space

**symbol0** · Jun22-09, 11:08 AM

Thanks Hurkyl