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Prove that if $f(x)$ has a greatest lower bound $L$ , then for every $\epsilon\gt 0$ there exists an $\bar{x}$ in the domain of $f(x)$ such that $|f(\bar{x})-L|\lt\epsilon$ .
Prove that if $f(x)$ is continuous on $[a,b]$ , then $f(x)$ takes on a minimum value on $[a,b]$ .
Prove the Extreme-Value Theorem.
Prove that if $f(x)$ has a least upper bound $M$ then for every $\epsilon\gt 0$ there exists an $\bar{x}$ in the domain of $f(x)$ such that $|f(\bar{x})-M|\lt\epsilon$ .
Prove that if $f(x)$ is continuous on $[a,b]$ , then $f(x)$ takes on a maximum value on $[a,b]$ .