Claim 7: The successive minima of a lattice are achieved i.e., for every $1 \leq i \leq n$, there exists a vector $v_i \in \Lambda$ with $\lVert v_{i}\rVert = \lambda_i(\Lambda)$.
There is a proof of that claim
Proof: By Corollary 6, the ball of radius (say) $2\lambda_i(\Lambda)$ contains only finitely many lattice points. It follows
from the definition of $\lambda_i$ that one of these vectors must have length $\lambda_i(\Lambda)$.
This suggests we should look at two places
Corollary 6: Let $\Lambda$
be a lattice. Then there exists some $\epsilon > 0$
such that $\lVert x − y\rVert > \epsilon$
for any two
non-equal lattice points $x, y \in\Lambda$.
Definition 7: Let $\Lambda$ be a lattice of rank $n$. For $i \in \{1, \dots , n\}$ we define the $i$th successive minimum as
$\lambda_i(\Lambda) = \inf \{r \mid \dim(\mathsf{span}(\Lambda ∩ \overline{B}(0, r))) \geq i\}$
where $\overline{B}(0, r) = \{x \in\mathbb{R}^m \mid \lVert x\rVert \leq r\}$ is the closed ball of radius $r$ around $0$.
How does this proof follow?
Consider $\overline{B}(0, \lambda_i(\Lambda))\cap \Lambda$.
By definition, this contains at least $i$ linearly independent lattice vectors (and is the smallest ball such that this occurs).
This is to say that shrinking the ball leads to a set containing at most $i-1$ linearly independent lattice vectors, i.e. the $i$th linearly independent lattice vector is on the surface of this ball.
Why does the claim state
the ball of radius (say) $2\lambda_i(Λ)$R contains only finitely many lattice points $\dots$
$\lambda_i(\Lambda)$ is defined with an infimum.
This is an "infinite" version of $\min$ (similarly to how $\sup$ is an "infinite" version of $\max$).
In particular, it is important to know that for a sequence $a_i$, $\inf a_i = a$ does not mean there is any particular index $i^*$ such that $a_{i^*} = a$.
For example, $a_i = i^{-1}$ (for $i > 0$) has an infimum of $0$ (as $i\mapsto 0^+$), but never achieves this limit.
This means that there was some risk that there is some infinite sequence of lattice points such that $\lVert \vec v_i\rVert\mapsto \lambda_i(\Lambda)$ in the limit, but there is no index $i$ such that this equality holds.
The proof handles this via noting that infimum (supremum) and min (max) agree on finite sets. Since we are restricting to a finite set, we can replace the $\inf$ with a $\min$, and note that it must be achieved by some element of the finite set.
