Proofs about this metric space In set X of all possible sequences $x=(n_1,n_2,...), y=(m_1,m_2,...)$ let $k_0(x,y)$ be the smallest index $k$ such that $n_k \neq m_k$. Prove:
a) $\rho(x,y) = k_0(x,y)^{-1}$  is a metric. DONE
b) triangle inequality is strengthened as $\rho(x,z) \leq$ max{$p(x,y), \rho(y,z)$}. DONE
c) for any open ball $B_r(x) = \bar{B}_r(x)$
d) $B_r(x)=B_r(y)$ for any $y \in B_r(x)$. DONE
e) if two balls have a common point then one of them is a subset of the other ball
I feel like this basically follows from d, but I can't figure out why exactly.
f) $x_n$ is a Cauchy sequence iff $\rho(x_n, x_{n+1}) \rightarrow 0$ as $n \rightarrow \infty$ Here my idea is to take a sequence that is Cauchy thenwe have $x_n \rightarrow x_{n+1}$ for n sufficiently large and as $n \rightarrow \infty, k_0(x_n,x_{n+1}) \rightarrow \infty$ so $\rho \rightarrow 0$ as $n \rightarrow \infty$.  For the other direction, I want to show $\rho(x_n,x_{n+1}) \rightarrow 0$ as $n \rightarrow \infty$  Then fix $\epsilon > 0$ then we want to find N such that $\forall m,n > N, \rho(x_n,x_m) < \epsilon$, but I'm having trouble fleshing out the details here.
g) X is complete and separable
To show X is complete, I think that we just need to show that every Cauchy sequence converges to a pt in X so by f we have that for any Cauchy seq, $\rho(x_n, x_m) \rightarrow 0$ so $x_n \rightarrow x_{n+1} \in X$.  I'm not sure thats right or how to show separable though
 A: For c), note that the metric takes discrete values, i.e. given any $a<b \in \mathbb{R}$ there is a value $c \in (a,b)$, which is never realized by any $\rho(x,y)$, since $k_0$ has integer values.
e) follows from d) as is explaned in the comment: if $z \in B_r(x) \cap B_r(y)$, then d) tells us that $B_r(x)=B_r(z)$ and $B_r(y)=B_r(z)$, so this is indeed just a weaker statement.
For f) and g), if you have a Cauchy sequence $(x_n)_n$, then $\forall \epsilon >0 ~\exists N ~\forall n,m\geq N: \rho(x_n,x_m)<\epsilon$. This means that $\forall K >0 ~\exists N ~\forall n,m\geq N:$ $x_n$ and $x_m$ have the same first $K$ entries. Moreover, in a general metric space, we have that if a subsequence of a Cuachy sequence converges, then the entire Cauchy sequence converges, as well. With this you should be able to prove f) and completeness in g).
For separability (I guess that the sequences are supposed to have integer entries), take the elements $x(n_1,\dots,n_k)=(n_1,\dots,n_k,0,0,0,\dots)$, where $n_i \in \mathbb{Z}$. These are countably many. Show that the collection of these elmenents is dense.
Cheers
