It looks like you are working on a proof involving sequences and limits. The statement you need to prove is:

Proposition: If $c$ is a nonzero real number and $(a_n)$ is a real-valued sequence that converges to $a$, then the sequence $(c a_n)$ converges to $c a$.

Steps to arrange:

1. Starting assumptions:

   • $(a_n)$ converges to $a$. That means, for any $\epsilon > 0$, there is an $N$ such that for all $n > N$, $|a_n - a| < \epsilon$.

2. What we need to show:

   • We need to show that $(c a_n)$ converges to $c a$. That is, for any $\epsilon > 0$, we want to find an $N$ such that for all $n > N$, $|c a_n - c a| < \epsilon$.

3. Use properties of limits:

   • Use the fact that $|c a_n - c a| = |c| \cdot |a_n - a|$, since $c$ is constant and nonzero. We want to show that for sufficiently large $n$, $|c| \cdot |a_n - a| < \epsilon$.

4. Choose $\epsilon$:

   • Let $\epsilon' = \frac{\epsilon}{|c|}$. Since $a_n \to a$, there exists an $N$ such that for all $n > N$, $|a_n - a| < \epsilon'$.