Can an Order-Preserving Injection Exist from w1 to the Reals?

[Dragonfall]

Homework Statement

Show that there does not exist an order-preserving injection from the ordinal $$\omega_1$$ to the reals (given the usual order).

The Attempt at a Solution

Suppose such an injection exists. Then something bad happens. Maybe the fact that w1 is well-ordered?

 

[morphism]

Give w_1 and R their order topologies. Then an order-preserving injection from w_1 into R is a topological embedding. But w_1 is not second countable, while R is. Contradiction, because a subspace of a second countable space is second countable.