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## Homework Statement

Prove [tex] 1 - x \leq e^{-x}[/tex] for [tex]0 \leq x \leq 1[/tex] by calculus.

## The Attempt at a Solution

Sketching shows that the statement seems to be true.

Let's assume that

[tex]f(x) = x + e^{-x} -1[/tex], and

[tex]f(x) \geq 0[/tex].

If the function is continuous and increasing, then the function gets all the

values between the interval [0, [tex]e^{-1}[/tex]], by Boltzman's Min-Max theorem.

Thus, the initial assumptions are true.

Please, point out any mistakes.