I've already proven uniform continuity of f(x)=sin(x) on R via epsilon-delta, but I wanted to try and prove it with the Heine-Cantor-Theorem since it seems more intuitive:
Now obviously with the theorem I get uniform continuity on any closed interval in R, e.g. [0;2π], but the problem I struggle with is then extending this on R. Since sin(x) is periodic, it seems obvious but I couldn't come up with a rigorous proof yet.
My best guess would be arguing with the fact I can "project" any two points "back" into a closed interval I already know sin(x) is uniformly continuous in, then "take" the delta from that interval and argue that this delta corresponds to the delta in the originating interval, but this idea doesn't seem very promising. Any help would be appreciated.