I'm helping a friend with his homework, and the following problem is really giving us both trouble.

Consider the subset $\displaystyle B$ of $\displaystyle \mathbb{R}^{2}$ given by $\displaystyle B=\left[\left([0,1]\times[0,3]\right)\cup\left([0,2]\times[0,2]\right)\cup\left([0,3]\times[0,1]\right)\right]\backslash\left\{\left(2,1\right)\right\}$. Give $\displaystyle B$ the partial order $\displaystyle \le'$ defined by $\displaystyle (a,b)\le'(c,d)$ if and only if $\displaystyle a\le c$ and $\displaystyle b\le d$, where $\displaystyle \le$ is the usual order on $\displaystyle \mathbb{R}$.

I'm good so far, but this is the part that has me stumped:

Find a set in $\displaystyle B$ which has no infimum.

It seems to me that any *finite* subset of $\displaystyle B$ would have an infimum.

(It should be noted that the book that this problem came from is well known for its typos and wrong answers in the student solutions section.)