A theorem due to Rosenthal is that every bounded sequence in a real or complex Banach space has either a weakly Cauchy subsequence or a subsequence that is equivalent to the standard $\ell_{1}$-basis. See the first page of this paper by Rosenthal.
If $(e_{i})_{i=1}^{\infty}$ is the standard basis for $\ell_{1}$ (over $\mathbb{R}$ for simplicity), then clearly, the above theorem implies that every subsequence is not weakly Cauchy. Take $(e_{i})_{i=1}^{\infty}$ as a subsequence of itself. Then, there is some $\varphi\in[\ell_{1}]^{*}$ and some $\epsilon>0$ such that \begin{equation} \epsilon\leq|\varphi(e_{i_{j}})-\varphi(e_{i_{k}})| \qquad j\neq k \end{equation} for some subsequence $(e_{i_{j}})_{j=1}^{\infty}$. On the other hand, it follows that $(\varphi(e_{i_{j}}))_{j=1}^{\infty}$ is a bounded sequence of real numbers since $|\varphi(e_{i_{j}})|\leq\|\varphi\|_{\text{op}}\|e_{i_{j}}\|_{1}=\|\varphi\|_{\text{op}}$ and so, has a convergent (and thus Cauchy) subsequence. Then, \begin{equation} \epsilon\leq|\varphi(e_{i_{j_{l}}})-\varphi(e_{i_{j_{m}}})| <\epsilon\end{equation} for $l,m$ large enough and this seems to be a contradiction. I know I have made a mistake somewhere because I seem to be asserting that every bounded sequence should have a weakly Cauchy subsequence and this is only true in spaces that do not contain a copy of $\ell_{1}$.
I cannot seem to figure out where my logic is wrong and any help is much appreciated!