Help with Metric Spaces! (Mathematics)

[titaniumx3]

Hi

I'm completely new to metric spaces and I'm having some difficulties getting around the notation and would appreciate some help. In particular I'm stuck on this:

What exactly does the part highlighted in red actually mean? I'm thinking the sup is 'supremum' but to what exactly does that apply? The largest value of |f(x)-g(x)| for any x in [0,1] ?

Also for part (a) I have no clue whatsoever! How would you show the sequence converges to 0?

Any advice would really be appreciated!

 

[titaniumx3]

Thanks that has helped quite a bit, but I still need to get my head around the the convergence of fn and how the distance affects it.

I understand that if fn is to converge to a function g, as n->inf then d(fn, g)->0. So this obviously means fn can only converge to g(x)=0 or g(x)=1, right?

 

[titaniumx3]

For this question you don't need to think about what it converges to. The question tells you that you are comparing the series of functions f_n to "the zero function." Convergence is all about distance. For one thing to converge to another the distance between them as n->inf must be 0.

Oh ok, that makes sense lol.

So in the case of d_1, as n->inf there always exists an x in [0,1] such that f_n(x)=1, hence where g(x)=0, d_1(f_n,g)=1, i.e. f does not converge to the zero function.

Hope thats on the right lines!

 

[titaniumx3]

Yeah, its more like a rectangle that gets progressively 'thinner'.

Thanks again to all for the help, I've actually begun to understand some of the lecture notes better now.

 

[titaniumx3]

That is very true and something I always try to do when approaching new definitions and theorems; just writing down a few basic examples of the theorem in practice helps a lot and as you say once you get a 'feel' for it the problems pretty much start to roll out themselves. The main issue I have though is that quite frequently it just takes rediculously long to actually comprehend the theorems/definitions and how exactly to apply them; don't even get me started on some of the proofs!