Let $f:[0,1] \to \mathbb R$ be a uniformly continuous function such that each value of $f(x)$ is greater than zero. Is its infimum greater than zero in BISH?

I believe that it is indeed the case if one assumes the Fan Theorem. But independent of it, I'm not sure.

[edit]

Note: It's possible to get around this problem in practice by interpreting $f > 0$ to mean that there exists a constant $c$ such that $f(x) > c > 0$ for all $x$. This is an example of a pseudo-order. The fact that the infimum is greater than zero is then a tautology.