Is this proof correct convergence I'm trying to show that $(a_{n}) \to -\infty \Leftrightarrow (-a_{n}) \to \infty$
Attempt
$(\Rightarrow)$: We have $(a_{n}) \to -\infty$. By definition this means, for all $C < 0$ there exists $N$ such that $a_{n} < C$ for all $n > N$. Therefore we have that $ a_{n} < C < 0$. Therefore for all $ -C > 0$ there exists $N$ such that $-a_{n} > -C > 0$ for all $n > N$
$(\Leftarrow$): We have $(-a_{n}) \to \infty$. By definition this means, for all $C > 0$ there exists $N$ such that $-a_{n} > C$ for all $n > N$. Therefore we have that $a_{n} < -C < 0$. Hence we have for all $-C < 0$ there exists $N$ such that $a_{n} < -C$ for all $n > N$.

Your proofs are correct. Well done.
