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_{n->∞}sup sin(n)=1, but I can only prove it using Diophantine approximation, and this question is for analysis. My teacher said we can do this, but it was obvious he prefer us do it another way that doesn't involve theorems we haven't learned, but I can't figure out another way to do this.

I was thinking of somehow showing that for any e>0, there are infinitely many terms of sin(n) such that sin(n)>1-e, but I am not even sure this is possible without other theorems not learned in analysis. I've also tried bounding it between things that converge to 1 at some point, but that was unsuccessful.

Can someone help?

Thanks.