Let the Minkowski metric $\eta_{\mu\nu}$ in $d+1$ space-time dimensions be
$$\eta_{\mu\nu}~=~{\rm diag}(1, -1, \ldots,-1).\tag{1}$$
Let the Lie group of Lorentz transformations be denoted as $O(1,d;\mathbb{R})=O(d,1;\mathbb{R})$. A Lorentz matrix $\Lambda$ satisfies (in matrix notation)
$$ \Lambda^t \eta \Lambda~=~ \eta. \tag{2}$$
Here the superscript "$t$" denotes matrix transposition. Note that the eq. (2) does not depend on whether we use east-coast or west-coast convention for the metric $\eta_{\mu\nu}$.
Let us decompose a Lorentz matrix $\Lambda$ into 4 blocks
$$ \Lambda ~=~ \left[\begin{array}{cc}a & b^t \cr c &R \end{array} \right],\tag{3}$$
where $a=\Lambda^0{}_0$ is a real number; $b$ and $c$ are real $d\times 1$ column vectors; and $R$ is a real $d\times d$ matrix.
Now define the set of orthochronous Lorentz transformations as
$$ O^{+}(1,d;\mathbb{R})~:=~\{\Lambda\in O(1,d;\mathbb{R}) | \Lambda^0{}_0 > 0 \}.\tag{4}$$
The proof that this is a subgroup can be deduced from the following string of exercises.
Exercise 1: Prove that
$$ |c|^2~:= ~c^t c~ = ~a^2 -1.\tag{5}$$
Exercise 2: Deduce that
$$ |a|~\geq~ 1.\tag{6}$$
Exercise 3: Use eq. (2) to prove that
$$ \Lambda \eta^{-1} \Lambda^t~=~ \eta^{-1}. \tag{7}$$
Exercise 4: Prove that
$$ |b|^2~:= ~b^t b~ = ~a^2 -1.\tag{8}$$
Next let us consider a product
$$ \Lambda_3~:=~\Lambda_1\Lambda_2\tag{9}$$
of two Lorentz matrices $\Lambda_1$ and $\Lambda_2$.
Exercise 5: Show that
$$ b_1\cdot c_2~:=~b_1^t c_2~=~a_3-a_1a_2.\tag{10}$$
Exercise 6: Prove the double inequality
$$ -\sqrt{a_1^2-1}\sqrt{a_2^2-1} ~\leq~ a_3-a_1a_2~\leq~ \sqrt{a_1^2-1}\sqrt{a_2^2-1},\tag{11}$$
which may compactly be written as
$$| a_3-a_1a_2|~\leq~\sqrt{a_1^2-1}\sqrt{a_2^2-1}.\tag{12}$$
Exercise 7: Deduce from the double inequality (11) that
$$ a_1\neq 0 ~\text{and}~ a_2\neq 0~\text{have same signs} \quad\Rightarrow\quad a_3>0. \tag{13}$$
$$ a_1 \neq 0~\text{and}~ a_2\neq 0~\text{have opposite signs} \quad\Rightarrow\quad a_3<0. \tag{14}$$
Exercise 8: Use eq. (13) to prove that $O^{+}(1,d;\mathbb{R})$ is stabile/closed under the multiplication map.
Exercise 9: Use eq. (14) to prove that $O^{+}(1,d;\mathbb{R})$ is stabile/closed under the inversion map.
The Exercises 1-9 show that the set $O^{+}(1,d;\mathbb{R})$ of orthochronous Lorentz transformations form a subgroup.$^{\dagger}$
References:
- S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; p. 57-58.
$^{\dagger}$A mathematician would probably say that eqs. (13) and (14) show that the map
$$O(1,d;\mathbb{R})\quad \stackrel{\Phi}{\longrightarrow}\quad
\{\pm 1\}~\cong~\mathbb{Z}_2\tag{15}$$
given by
$$\Phi(\Lambda)~:=~{\rm sgn}(\Lambda^0{}_0)\tag{16}$$
is a group homomorphism between the Lorentz group $O(1,d;\mathbb{R})$ and the cyclic group $\mathbb{Z}_2$, and a kernel
$$ {\rm ker}(\Phi)~:=~\Phi^{-1}(1)~=~O^{+}(1,d;\mathbb{R}) \tag{17}$$
is always a normal subgroup.
For a generalization to indefinite orthogonal groups $O(p,q;\mathbb{R})$, see this Phys.SE post.
