minimum_maximum_theorem

introduction

%%visits: 2 ## intuition extreme_value_theore := Maximum/Minimum theore := If f ∈ C[a, b] then f attains its maximum and minimum

Proof. Denote M = sup(f).

$\forall n \in \mathbb{N} \exists x_n \in [a,b] : \left| f(x_n) - M \right| < \frac{1}{n}$. By Bolzano-Weierstrass theorem, ∃x_nk → c as k → ∞. f is continous, so f(x_nk) → f(c)ask → ∞. Thus M = f(c), The proof for minimum is the similar, with inf f

proof

Choose X_N so f(X_N) → M = supf(t)

$M \ge f(X_N) > M - \frac{1}{N}$

B-W X_NI → p

Triangle inequality $| f(p) - M | | f(p) - f(X_N_I) | + | f(x_N} | $ ## rigour ## exam clinic ## examples and non-examples ## resources tags :math:real_analysis:

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