Let $A\in\mathbb{R}^{m\times n}, \ c\in \mathbb{R}^n, \ b\in\mathbb{R}^m$ and consider the linear program $\max \{ c^Tx : Ax\le b\} \ (1)$ Its dual is $\min \{ b^Ty : A^Ty=c, \ y\ge 0\} \ (2)$ The weak duality theorem states that for $x$ feasible for (1) and $y$ feasible for (2), then $c^tx\le b^ty$
The following statement is obviously false, but where is the flaw ?
It has been shown that the dual of (2) is equivalent to (1). Then if $x$ is feasible for (1) and $y$ is feasible for (2), we have $c^tx\le b^ty$ by the theorem applied to (1) and its dual (2) and $b^ty\le c^tx$ by the theorem applied to (2) and its dual (1). Hence $c^tx=b^ty$ for any $x$ and $y$ like above.