latex 大于等于号

按要求geq 后有空格，但是还是报错

egin{thm}label{thm1}
    (Coles et al., 2001)). Assume that Theorem 1 holds with $xi$ 	extless 0 and let the upper endpoint of F be denoted by $b^ *$. Then, \
    P$igg(frac{M_n-a_n}{b_n} leq zigg)$$
ightarrow$W(z),   qquad       n$
ightarrow$+$infty$,\
    where\
    $$W(z)=
    egin{cases}
    exp iggl{-igg(frac{b^*-z}{sigma} igg)^alpha  iggr},  & if z	ext{	extless$b^*$},\
    1& if z	ext{$geq $$b^*$ },\此行老是报错
    end{cases}$$
    
end{thm} 

后来，改了在公式环境中进行，在geq 前后加入了$$,报错消除

egin{thm}label{thm1}
    (Coles et al., 2001)). Assume that Theorem 1 holds with $xi$ 	extless 0 and let the upper endpoint of F be denoted by $b^ *$. Then, \
    P$igg(frac{M_n-a_n}{b_n} leq zigg)$$
ightarrow$W(z),   qquad       n$
ightarrow$+$infty$,\
    where\
    $$W(z)=
    egin{cases}
    exp iggl{-igg(frac{b^*-z}{sigma} igg)^alpha  iggr},  & if z	ext{	extless$b^*$},\
    1& if z	ext{$geq $$b^*$ },\
    end{cases}$$
    
end{thm} 

其中得到的公式如下