I have tried to make sense of the question by reversing four of the inequality signs. I have also replaced i by x, partly because i tends to mean the square root of –1 and partly because it's easier to prove the result if you think of x as a variable and t as a constant.

The Mean Value theorem (with remainder) for the function tells you that for some y between 0 and x. Since , it follows that .

(If you don't know the Mean Value theorem, we'll have to think of another approach.)