Boundedness Theorem  


If f is continuous on [a,b], then f is bounded on [a,b]


Consider it

E := {x|x∈[a,b]∧f is bounded on [a,x]} (but, b is one of the upper bound on E)


and we can choose c that

c = lub {E} ∈ ℝ (Or sup{E} ∈ ℝ)


we argue that c = b. first we suppose c<b, From the continuity of f at c, (and ∃ε > 0) so, f is bounded on [c - ε, c + ε], it is obviously

f is bounded on [a, c + ε]. This is contradiction(∵ c - ε will be upper bound on E, then c - ε is l.u.b)


so c = b. this tell us f os bounded on [a, x] for all x < b. we know from f is continuity f is bounded on some closed interval [b-ε, b] (and b-ε < b)

so, It is obviously f is bounded in [a, b-ε]∧[b-ε, b]


∴ if f is continuous on [a, b], then f is bounded on [a, b]


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