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**Jose27** A hint: $\displaystyle X$ is reflexive so $\displaystyle K$ has weakly compact (weak) closure, in particular any sequence $\displaystyle (x^{(n)})$ in $\displaystyle K$ has a (weak) limit point say $\displaystyle x$ , use this to show that for $\displaystyle k\in \mathbb{N}$ you can have $\displaystyle \sum_{i=1}^{k} |x^{(n)}_i-x_i| <\varepsilon$ for $\displaystyle n$ big enough. Now use your last condition to show that we can also control the tails uniformly in $\displaystyle n$.