1. ## Bounded spaces

I have proved that a totally bounded metric space is bounded using the idea of epsilon nets, but can't work the attached out.

Any solutions would be great thanks.

2. Originally Posted by Cairo
I have proved that a totally bounded metric space is bounded using the idea of epsilon nets, but can't work the attached out.

Any solutions would be great thanks.
Notice that $\displaystyle d(e_k,e_l)=\sqrt{2}$ if $\displaystyle k\neq l$ now try to find a $\displaystyle \epsilon$-net with $\displaystyle \epsilon < \sqrt{2}$. That it's bounded should be trivial.

3. Where did you get root(2) from? I thought d(a,b) was root(1)????

4. Originally Posted by Cairo
Where did you get root(2) from? I thought d(a,b) was root(1)????
If $\displaystyle k\neq l$ then $\displaystyle d(e_k,e_l)= \left( \sum_{n=1}^{\infty } |e_{k}^n - e_{l}^n|^2 \right) ^{\frac{1}{2} } = (1+1)^{\frac{1}{2} }$ because $\displaystyle e_k$ and $\displaystyle e_l$ have their respective $\displaystyle 1$ in different places.

Also it doens't matter which number it is, just that each point is a constant distance away from each other, and so any net we try to build with a number smaller than this constant will fail to cover all from a finite point set.

5. Okay....but i'm still not sure how the proof would go.

Sorry for being thick here!

6. Originally Posted by Cairo
Okay....but i'm still not sure how the proof would go.

Sorry for being thick here!
Assume it is totally bounded and pick a number $\displaystyle 0<a<\sqrt{2}$ then we must have that there exist $\displaystyle e_{m_1},...,e_{m_k} \in (e_j)_{j\in \mathbb{N} }$ such that $\displaystyle (e_j)_{j\in \mathbb{N} } \subseteq \cup_{i=1}^{k} B_a(e_{m_i} )$ but $\displaystyle B_a(e_{m_i} ) = \{ y \in l_2 : d(y,e_{m_i})<a \}$ and so $\displaystyle (e_j)_{j\in \mathbb{N} } \cap B_a(e_{m_i} ) = \{ e_{m_i} \}$