Prove that for all \alpha, \beta in \omega_1,

 \text{seg}_{\omega_1}(\alpha) +_o \text{seg}_{\omega_1}(\beta) <_o \omega_1

and

<br />
\text{seg}_{\omega_1}(\alpha) \cdot_o \text{seg}_{\omega_1}(\beta) <_o \omega_1.

Notation: \omega_1 denotes the first uncountable ordinal. \text{seg} above denotes the initial segment, \cdot_o denotes the product of posets, +_o denotes the sum of posets. U <_o V \Leftrightarrow (\exists x \in V ) [U =_o \text{seg}_V(x)]. =_o denotes order isomorphic.