1. ## inequalities

Use the inequality absolute value[sin(a)-sin(b)] << absolute value[a-b], which is valid for all real numbers a and b, to prove that sin x is continuous on R.

2. just apply the mean value theorem for the function $f(t)=\sin t$ on the interval $[a,b].$

3. Is the mean value theorem the same thing as the intermediate value theorem?

IVT
f(x) is defined on [a,b]
f(x) assumes all values of (a,b)
then f(c) where c is a point in (a,b) intersects f(x)

4. no, the assumptions are different, google it and see.