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continuous function

How do I prove this ?

Let [a,b] be a closed and bounded real interval. Prove that a continuous function f : [a,b] -> R is bounded.
Reply 1
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B_9710
18
Have you not been given any hints or anything to be able to do this? To come up with a proof from scratch seems difficult.
Original post by B_9710
Have you not been given any hints or anything to be able to do this? To come up with a proof from scratch seems difficult.


well the previous part of the question says to state the bolzano-weirerstrass theorem so i assume you use that as a hint
anyone plz
sad times
Reply 5
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B_9710
18
If it's unbounded (assume not bounded above) then there exists some sequence xn x_n such that f(xn)>n f(x_n)>n . Now xn∈[a,b] x_n \in [a, b] which is bounded so has a convergent subsequence that convereges to x say. You can use the fact that f is sequentially continuous at x as it's continuous at x. You can try and find a contradiction which means that f I see bounded above.

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