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Response Details:
After I posted my result I looked at the previous answer. Somehow the behaviour of f near 0, which is key to solving the question, disappeared from his answer completely, and rather mysteriously. Also, the inequality in the second line is wrong. If you look at the series expansion for e^h - 1, you get h + h^2/2 + h^3/3! + ..., so if you take a positive h, how is this supposed to be <= h? h is just the first term, and you are adding all those other positive terms as well!
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