closed embedded submanifold and proper map

Question

In Lee's Introduction to Topological Manifolds. I came across this statement:

A submanifold of $M$ is closed in $M$ if and only if the inclusion map is proper.

However, isn't the inclusion map from $i: B^k \to R^k$ a counter example to this, since the map is an embedding and proper but $B^k$ is open in $R^k$.

Answer 1

The inclusion map is not proper, because preimage of $\bar{B(1,0.5)}$is not closed, therefore not compact