latex 大于等于号

按要求\geq 后有空格，但是还是报错

\begin{thm}\label{thm1}
    (Coles et al., 2001)). Assume that Theorem 1 holds with $\xi$ \textless 0 and let the upper endpoint of F be denoted by $b^ *$. Then, \\
    P$\bigg(\frac{M_n-a_n}{b_n} \leq z\bigg)$$\rightarrow$W(z),   \qquad       n$\rightarrow$+$\infty$,\\
    where\\
    $$W(z)=
    \begin{cases}
    exp \biggl\{-\bigg(\frac{b^*-z}{\sigma} \bigg)^\alpha  \biggr\},  & if z\text{\textless$b^*$},\\
    1& if z\text{$\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

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

其中得到的公式如下