Question: I'm studying Probability and can't solve this question:

Random tournaments: consider a sport tournament in which n teams
participate. Each team plays one match against each other team. Each such
match ends with one of the two teams winning (there are no draws). Assume
that all teams are equally strong, so each match has a random winner. We say
that a team is the clear winner of the tournament if it won all its n − 1 matches
(this team beats all other teams). What is the probability that the tournament
ends without a clear winner for n = 10?

Answer: 
Because $h$ is continuous on $K,$ so is $|h|.$ Hence $|h|$ assumes a minimum value $m$ at some $x\_0\in K.$ Because $|h(x\_0)|>0,$ we have $m> 0.$ But notice $|h| \ge m \implies \frac{1}{|h|} \le \frac{1}{m} ,$ and we're done.
If it is unbounded, then there is a sequence $(x\_n) \subset K$ such that $\pm \infty$ is a cluster point of $1/f(x\_n)$. Hence, $0$ is a cluster point of $f(x\_n)$, so $(x\_n)$ has a subsequence $(y\_n)$ such that $f(y\_n) \to 0$. By the compactness of $K$, $(y\_n)$ has a convergent to some $z \in K$ subsequence $(z\_n)$, and by the continuity of $f$, $f(z) = \lim f(z\_n) = 0$, $(f(z\_n))$ being a subsequence of $(f(y\_n))$. A contradiction.