How can any supremum of a proper subset of the real numbers be outside the set, following the logic below?

Let E ⊂ R, E is bounded above, and α = sup E. Then ∀ε> 0, there exists some t ∈ E such that

α−ε< t ⇒α− t < ε

Since α> t, α− t = |α− t|. It follows that

|α− t| < ε

which implies α = t ∈ E.

Why is this wrong?