Problem: Let $c_0$ be the space of complex sequences that converge to $0$. Let $\{ y_n\}_{n \in \mathbb{N}} $ be a sequence of scalars such that for arbitrary sequence $\{ z_n\} \in c_0 $ we have: $ \{ y_n z_n\} \in c_0 $. Prove that $\{ y_n\}_{n \in \mathbb{N}} \in \ell_{\infty}$ . "
I know this can be proved using Uniform boundedness theorem, but I'm interested in proof that uses $ ( p \implies q ) \iff ( \lnot q \implies \lnot p) $. So, the following Statement (*) should be proved:
"If $\{ y_n\}_{n \in \mathbb{N}} \notin \ell_{\infty} $, then there exists a sequence $\{ z_n\}_{n \in \mathbb{N}} \in c_0 $ such that $\{y_n z_n\}_{n \in \mathbb{N}} \notin c_0 $. " $(*)$
My attempt: Assume that $\{ y_n\}_{n \in \mathbb{N}} \notin \ell_{\infty}$. Hence, there is a subsequence $\{ y_{n_k} \}_{k \in \mathbb{N}} $ which is unbounded i.e. $ \lim_{ k\rightarrow \infty} | y_{n_k}| = + \infty. $ Now, let $ z_{n_k} = \frac{1}{y_{n_k} } $. Then $\{ z_{n_k} \}_{{k \in \mathbb{N}} }$ belongs to $ c_0$. Can we proceed by evaluating limit
$ \lim_{k \rightarrow \infty } | y_k z_{n_k} | $ and try to show it is not $0$? Can the statement ($*$) be proved like this?
Thanks a lot in advance.