TITLE: How to prove a manifold is diffeomorphic to Euclidean space?
QUESTION [9 upvotes]: Problem is this: suppose a manifold 
$$M=\bigcup_{n\in\mathbb{N}} U_n,$$ 
where each $U_n$ is diffeomorphic to Euclidean space, and $U_n$ is contained in $U_{n+1}$. Then please show that $M$ is diffeomorphic to Euclidean space.

REPLY [2 votes]: Here is a proof which works for $m=dim(M)\ne 4$, I am not sure about the remaining case $m=4$. 
By shrinking each $U_n$ we obtain an exhaustion of $M$ by nested smooth closed $m$-dimensional balls:
$$
B_1\subset B_2 \subset ... 
$$
By the h-cobordism theorem, for each $n$, the region $B_n- int(B_{n-1})$ is diffeomorphic to the annulus $S^{m-1}\times [0,1]$. By lining up these diffeomorphisms, we obtain a diffeomorphism $M- int(B_1)\to S^{m-1}\times [0,1)$. 
What makes this "lining up" work is the fact that if $A$ is a manifold with boundary diffeomorphic to $S^k\times [0,1]$ and $h$ is a diffeomorphism from one boundary component of $A$ to $S^k\times \{0\}$, then $h$ extends to a diffeomorphism $A\to S^k\times [0,1]$. 
Hence, $M$ is diffeomorphic to $R^m$. 
In the topological category, this argument also shows that $M$ is homeomorphic to $R^m$ even if $m=4$. (Since the topological annulus conjecture is proven by Quinn in dimension 4.)