I'm reading toric variety from Cox et. al. and there is a theorem about proper mapping that I cant solve.
Let $f : X \rightarrow Y$ be a continuous mapping of locally compact first countable Hausdorff spaces. Then the following are equivalent:
(a) $f$ is proper
(b)Every sequence $(x_k)\in X$ such that $f(x_k)\in Y$ converges in $Y$ has a subsequence $(x_{k_l})$ which converges in $X$.
This is the definition of Proper map.
A continuous mapping $f : X \rightarrow Y$ is proper if the inverse image $f^{-1}(T)$ is compact in $X$ for every compact subset $T \subseteq Y$. It will great if you can give me some hints about how to solve it!
(a) implies (b) is clear.
(b) implies (a): Recall that a subset $S\subset X$ is compact iff for every sequence $x_n$ in $S$, there is a convergent subsequence.
Let $T$ be a compact subset of $Y$. Let $S=f^{-1}(T)$. Given a sequence $x_n\in S$, $f(x_n)=y_n$ in $T$ has a convergent subsequence say $f(x_{n_k})=y_{n_k}$.
Now apply the condition (b) to the sequence $x_{n_k}$, so we get this one has a subsequence $x_{n_{k_r}}$ which is convergent in $X$ hence in $S$. So we are done.