A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field.
I want to show that weakly Cauchy sequences are not necessarily weakly convergent.
This seems to be the case for Hilbert spaces On the limits of weakly convergent subsequences, whats the difference?
In the link, it's actually true for a reflexive space.
So to get a counter-example, we have to think on a non-reflexive Banach space.
Take $X:=C[0,1]$ endowed with the supremum norm and $x_n(t):=t^n$. This sequence doesn't converge weakly, but for all $l\in X'$, we can represent it by a Radon measure, and $\left\{\int_{[0,1]}t^d\mu(t)\right\}$ is convergent by monotone convergence theorem applied to the positive and negative parts of $\mu$.