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I was reading Wiki, I met a problem in understanding the the proof of boundedness theorem exactly when they said

"Because [

but Bolzano theorem state that if the sequence is bounded, which is not necessary in our case.

What I miss here

And in the alternative proof they said

"The set {

f is continuous at [a,b] but how should it be bounded it is clear but how to prove that ?

Thanks

"Because [

*a*,*b*] is bounded, the Bolzano–Weierstrass theorem implies that there exists a convergent subsequence"but Bolzano theorem state that if the sequence is bounded, which is not necessary in our case.

What I miss here

And in the alternative proof they said

"The set {

*y*∈**R**:*y*= f(*x*) for some*x*∈ [*a*,*b*]} is a bounded set."f is continuous at [a,b] but how should it be bounded it is clear but how to prove that ?

Thanks

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