proof that $\omega$ has the tree property

Let $T$ be a tree with finite levels and an infinite number of elements. Then consider the elements of $T_0$. $T$ can be partitioned into the set of descendants of each of these elements, and since any finite partition of an infinite set has at least one infinite partition, some element $x_0$ in $T_0$ has an infinite number of descendants. The same procedure can be applied to the children of $x_0$ to give an element $x_1\in T_1$ which has an infinite number of descendants, and then to the children of $x_1$, and so on. This gives a sequence $X=⟨x_0,x_1,\mathrm{\dots }⟩$. The sequence is infinite since each element has an infinite number of descendants, and since $x_{i+1}$ is always of child of $x_i$, $X$ is a branch, and therefore an infinite branch of $T$.