TITLE: Why König's lemma isn't "obvious"?
QUESTION [25 upvotes]: I keep facing König's lemma "Every finitely branching infinite tree over $\mathbb{N}$ has infinite branch". Why it is not taken "obvious" but needs a careful proof? 
It seems somewhat obvious, but I guess I overlook something.

REPLY [8 votes]: It might be instructive to see that Kőnig's lemma is false (and why it fails) in a computable setting.  See this very clear exposition by Andrej Bauer and in particular theorem 3.5.