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## Homework Statement

Suppose the function ##f:[0,1] \to \mathbb{R}## is continuous, ##f(0) > 0## and ##f(1)=0##. Prove that there is a number ## x_0 \in (0,1] : f(x_0) = 0## and ##f(x) > 0## for ##0 \leq x \leq x_0##.

## Homework Equations

We can't use the IVT. Additionally, the definition of continuity we have been given is, a function ##f: D \to \mathbb{R}## is continuous at ##x_0## in ##D## if whenever ##\{x_n \}## is a sequence in ##D## that converges to ##x_0## the image sequence ##\{ f(x_n) \}## converges to ##f(x_0)##.

## The Attempt at a Solution

I was thinking about considering the set ##S=\{f([0,1]) \}## and using the Completeness Axiom to label the ##\inf S## then using the Bolzano-Weierstrass Theorem as part of a constructive existence proof.

Can you tell me if I'm on the right track, or is there a better way to start?

Thanks a ton!