I've been reading about reverse mathematics (mostly on wikipedia), and I had been thinking that I understood how to prove the equivalences to WKL0 and ACA0 mentioned in the its article. However, I now realize that my idea of how WKL0 can prove that every continuous real valued function on [0,1] is bounded. My idea would have started "since f is continuous, there f is locally bounded near each point, so there is an open cover of [0,1] such that f is bounded on each member of the cover", but I can't figure out how to express "f is bounded on the interval with rational endpoints (q,r)" as a Sigma_1 property, and I can't figure out how to get around this issue, either.

How does WKL0 prove that every continuous real valued function on [0,1] is bounded?