Suppose we have the following minimization problem:

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

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