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Let $X$ and $Y$ be topological spaces. A continuous map $F:X \rightarrow Y$ is called proper if the preimage of any compact subset in $Y$ is a compact subset of $X$. I wish to understand the definition of a proper map because it is a key ingredient in studying when a quotient of a smooth manifold is again a smooth manifold.

We say that sequence $(p_i)_{i=1}^\infty$ in $X$ escapes to infinity if each compact subset of $X$ contains finitely many $p_i$.

On the wikipedia page for proper maps, we find the following unreferenced claim:

Proposition. Let $X$ and $Y$ be metric spaces, and let $F:X \rightarrow Y$ be a continuous map. Then $F$ is proper if and only if $(F(p_i))_{i=1}^\infty$ escapes to infinity whenever $(p_i)_{i=1}^\infty$ is a sequence in $X$ which escapes to infinity.

I believe that I have proved the forward direction: Suppose $F$ is proper. Let $(p_i)_{i=1}^\infty$ is a sequence in $X$ which escapes to infinity, and set $A := \{p_i : i = 1,2,\ldots\}$. Let $K \subseteq Y$ be compact. Then $F^{-1}(K)$ is compact, and so $F^{-1}(K) \cap A$ is a finite set. Thus, $F(F^{-1}(K) \cap A) = K \cap F(A)$ is a finite, set, so $(F(p_i))_{i=1}^\infty$ escapes to infinity.

I am having trouble showing the other direction. I know that compactness in a metric space is equivalent to sequential compactness, but I am not sure how to use this. Can I get a hint?

Joseph Kwong
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Let $K\subset Y$ be compact. Take a sequence $(q_i)\in F^{-1}(K)\subset X .$ By compactness there's a convergent subsequence, $F(q_{i_k})\in K.$ Let $p_i:=F(q_i).$

Hence $(p_i)_{i=1}^\infty $ doesn't escape to infinity in $Y.$

So...there's a compact set $C\subset X$ with infinitely many $q_i\in C\subset X.$ By compactness it has a convergent subsequence. But this subsequence lies in $F^{-1}(K).$

suckling pig
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