Understanding a Proof (Confidence Intervals)

iocal

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Hi all, I need some help understanding the following proof. The argument is a bit dense so please bear with me.

Theorem
Let \(\displaystyle X_1.X_2,...,X_n\) be a random sample on a discrete random variable X with pmf p(x;\(\displaystyle \theta\)), where \(\displaystyle \theta\) is the parameter we wish to estimate. Let \(\displaystyle T=T(X_1,X_2,...,X_n)\) be an esimator of \(\displaystyle \theta\) with cumulative distribution function \(\displaystyle F_T(t;\theta)\). Assume that \(\displaystyle F_T(t;\theta)\) is a decreasing function of \(\displaystyle \theta\) for every t in the support of T. Let \(\displaystyle a_1>0\) and \(\displaystyle a_2>0\) be given and let \(\displaystyle \underline{\theta}\) and \(\displaystyle \bar{\theta}\) be solutions of the equations

\(\displaystyle \displaystyle F_T(T-;\underline{\theta})=1-a_2\) and \(\displaystyle \displaystyle F_T(T;\bar{\theta})=a_1\)

Where \(\displaystyle T-\) is the statistic whose support lags by one value of T's support. Under these conditions the interval(\(\displaystyle \underline{\theta},\bar{\theta}\)) is a confidence interval for \(\displaystyle \theta\) of at least \(\displaystyle 1-a_1-a_2\)

Proof
Define

\(\displaystyle \displaystyle\bar{\theta}=sup\left\{\theta:F_T(T; \theta)\geq a_1\right\}\)
\(\displaystyle \displaystyle\underline{\theta}=inf\left\{\theta: F_T(T-;\theta)\leq 1-a_2\right\}\)

Hence

\(\displaystyle \displaystyle \theta>\bar{\theta}\Rightarrow F_T(T;\theta)\leq a_1\)
\(\displaystyle \displaystyle \theta<\underline{\theta}\Rightarrow F_T(T-;\theta)\geq1-a_2.\)

These implications lead to
\(\displaystyle \displaystyle P[\underline{\theta}<\theta<\bar{\theta}]=1-P[\theta<\underline{\theta}]-P[\theta>\bar{\theta}]\geq 1-P[F_T(T-;\theta)\geq 1-a_2]-P[F_T(T;\theta)\leq a_1]\geq 1-a_1-a_2\)

What I do not understand is where the last inequality comes from, i.e. \(\displaystyle 1-a_1-a_2\). The book says that it is evident from the supremum and infimum equations above but apparently it is not evident for me, far from it. I would not normally post something that long but I've been stuck here for days so I am a bit desperate. Please help!
 
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