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.
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.
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 $\displaystyle (e_j)_{j\in \mathbb{N} } \cap B_a(e_{m_i} ) = \{ e_{m_i} \}$