## Minimization problem

Suppose we have the following minimization problem:

$\text{minimize}_{z \in \mathbb{R}^{N}} \sum_{i=1}^{N} \frac{|z_i|}{(|z_{n,i}|+ \varepsilon_n)^{1-q}}$ subject to $A \bold{z} = \bold{y}$. How do we prove the following:

Proposition. For any nonincreasing sequence $(\varepsilon_n)$ of positive numbers and for any initial vector $\bold{z}_0$ satisfying $A \bold{z}_0 = \bold{y}$, the sequence above admits a convergent subsequence?