$\text{Let}$$$\lim_{x\to d}h(x)=P,$$$$\lim_{x\to d}k(x)=Q\text{, and}$$$$h(x)\geq k(x)\text{ for all }x\text{ in an open interval containing }d.$$ $\text{Show}$$$P\geq Q$$
$\text{This is what I tried to do:}$$$|h(x)-P|\leq\epsilon,$$$$|k(x)-Q|\leq\epsilon,$$ $$-\epsilon\leq h(x)-P\leq\epsilon$$ $$-\epsilon\leq k(x)-Q\leq\epsilon$$ $$0\leq h(x)-k(x)-P+Q\leq0$$ $$|h(x)-k(x)-P+Q|=0\leq|h(x)-k(x)|+|P-Q|$$ $\text{But I get stuck. What am I doing wrong?}$