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The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which fits into a few lines, is accessible to students with limited calculus experience, and does not involve too much analysis of different cases.

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Question about the same inequality: Prove that $e^x\ge x+1$ for all real $x$ –  Martin Sleziak Sep 25 '13 at 12:31
    
There are many great answers below. I now wonder whether, as a question with no objective right answer, this should be community wiki? –  Ashley Montanaro Sep 26 '13 at 21:47

18 Answers 18

Another way (not sure if its "simple" though!): $y = x+1$ is the tangent line to $y = e^x$ when $x= 0$. Since $e^x$ is convex, it always remains above its tangent lines.

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That's my favourite argument. I'm afraid it fails on the "is accessible to students with limited calculus experience" criterion, but it's boootiful. –  Daniel Fischer Sep 25 '13 at 12:28
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@DanielFischer: Are you sure that it fails? Tangent lines can appear much earlier then the derivatives and it have a nice graphical representation (sure, representation might not be mathematically precise). –  Maciej Piechotka Sep 25 '13 at 13:38
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@Maciej I'm not sure. But without some calculus to build on, I think it would be too hand-wavy for me. –  Daniel Fischer Sep 25 '13 at 13:44
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The convexity, at least, is elementary: $e^{x+z} - 2 e^x + e^{x-z} = (e^z-1)^2 e^{x-z} > 0$ for all real $x,z$, with equality iff $z=0$. –  Noam D. Elkies Sep 25 '13 at 14:55
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@NoamD.Elkies -- for large values of "elementary". –  Malvolio Sep 25 '13 at 18:52

The shortest proof I could think of: $$1 + x \leq 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = e^x.$$

However, it is not completely obvious for negative $x$.

Using derivatives:

Take $f(x) = e^x - 1 - x$. Then $f'(x) = e^x - 1$ with $f'(x) = 0$ if and only if $x = 0$. But this is a minimum (global in this case) since $f''(0) = 1 > 0$ (the second derivative test). So $f(x) \geq 0$ for all real $x$, and the result follows.

Another fairly simple proof (but it uses Newton's generalization of the Binomial Theorem which is often covered in precalculus):

We proceed by contradiction. Suppose the inequality does not hold, i.e., $e^x < 1 + x$ for some $x$. Then $e^{kx} < (1 + x)^k$. Now set $x = 1/k$ so that \begin{align*} e &< \left( 1 + \frac{1}{k} \right)^k\\ &= 1 + \frac{k}{1}\left( \frac{1}{k} \right)^1 + \frac{k(k - 1)}{1 \cdot 2}\left( \frac{1}{k} \right)^2 + \frac{k(k - 1)(k - 2)}{1 \cdot 2 \cdot 3}\left( \frac{1}{k} \right)^3 + \cdots\\ &< 1 + \frac{k}{1}\left( \frac{1}{k} \right)^1 + \frac{k^2}{1 \cdot 2}\left( \frac{1}{k} \right)^2 + \frac{k^3}{1 \cdot 2 \cdot 3}\left( \frac{1}{k} \right)^3 + \cdots\\ &= 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots\\ &= e, \end{align*} which is absurd. Therefore $1 + x \leq e^x$ for all real $x$.

By the way, this is where $$e = \lim_{k \to \infty}\left( 1 + \frac{1}{k} \right)^k$$ comes from because $$\lim_{k \to \infty}\frac{k(k - 1)}{k^2} = \lim_{k \to \infty}\frac{k(k - 1)(k - 2)}{k^3} = \cdots = \lim_{k \to \infty}\frac{k(k - 1)(k - 2) \cdots (k - n)}{k^{n + 1}} = \cdots = 1.$$

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Is it that evident for negative $x$? –  Macavity Sep 25 '13 at 10:16
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@Macavity for x<-1 is obvious –  BЈовић Sep 25 '13 at 17:35
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In your third proof : Why should such a $x$ be of the form $1/k$ with $k$ a positive integer ? –  user37238 Sep 26 '13 at 15:20
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@user37238 Here, $k$ is not necessarily an integer because, technically, the proof uses the Binomial Series. This explains why I wrote the binomial coefficients using the falling factorial. –  glebovg Sep 26 '13 at 17:58

Let $f(x) = e^x-(1+x)$, then $f^\prime(x) = e^x-1$. Hence $f^\prime(x)=0$ iff $x=0$. Furthermore $f^{\prime\prime}(0) = e^0=1>0$, thus $f(0)=0$ must be the global minimum of $f$, proving your claim.

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I like the idea behind this proof : just study a function and show that this function is always non-negative. But I think that it could be a little simpler if you say that $f'\ge 0$ on $[0,+\infty($ (and consequently $f$ increases on this interval) and $f'\le 0$ on $)-\infty,0]$ (and $f$ decreases on this interval) so $f(0)=0$ is a global minimum of $f$. –  user37238 Sep 25 '13 at 14:59

$$ e^x = \lim_{n\to\infty}\left(1+\frac xn\right)^n\ge1+x $$

by Bernoulli's inequality.

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This is what I thought of when I saw this question. (+1) –  robjohn Sep 27 '13 at 0:31

One that's not been mentioned so far(?): knowing that $$ 0 < e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots $$ proves the inequality except for $-1 < x < 0$. But in that region $$ e^x - (1+x) = \frac{x^2}{2} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots $$ is an alternating series whose terms decrease in absolute value and start out positive. Therefore it is positive by the usual argument: group the terms as $$ e^x - (1+x) = \left( \frac{x^2}{2} + \frac{x^3}{3!} \right) + \left( \frac{x^4}{4!} + \frac{x^5}{5!} \right) + \cdots $$ and observe that each combined term is positive, QED.

(This actually works for $-3 < x < 0$, but you still want to use $e^x > 0$ to prove the inequality for very negative $x$.)

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Why does knowing that $0<e^x$ give the result for $x\le-1$? –  Donkey_2009 Sep 25 '13 at 14:56
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$x \leq -1$ means $x+1 \leq 0 < e^x$. –  Noam D. Elkies Sep 25 '13 at 14:58
    
What is $n$? ${}{}{}{}$ –  user37238 Sep 25 '13 at 15:22
    
Good question: it's a typo and should be $x$. I've fixed it now. Thank you. –  Noam D. Elkies Sep 26 '13 at 20:51

If $x \ge 0$ then $$\begin{align} e^x = 1 + \int_0^x e^t\,\mathrm dt &= 1 + \int_0^x\left( 1 + \int_0^t e^u \,\mathrm du\right)\,\mathrm dt \\&= 1+x + \int_0^x \int_0^t e^u\,\mathrm du\,\mathrm dt \ge 1+x\end{align}$$

If $x \le 0$ then $$\begin{align}e^x = 1 - \int_x^0 e^t\,\mathrm dt &= 1 - \int_x^0\left( 1 - \int_t^0 e^u\,\mathrm du\right)\,\mathrm dt \\ &= 1+x + \int_x^0 \int_t^0 e^u\,\mathrm du\,\mathrm dt \ge 1+x\end{align}$$

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For completeness, using $exp(x)=1+x+\frac{1}{2}x^2+\dots$, the inequality is trivial for $x\ge 0$. It is also trivial for $x<-1$.

It remains to show the case $-1<x<0$. Replacing $x$ by $-x$, one need to show $1-x < e^{-x}$ for $0<x<1$, or $$1+x+\frac{1}{2}x^2+\dots=e^x <\frac{1}{1-x}=1+x+x^2+\dots,$$ we are done.

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We want to prove that $1+x\le e^x$ for any $x\in\mathbb R$. Setting $x=\log(u)$, this is equivalent to proving:

$$ 1+\log(u)\le u $$ for any $u\in (0, \infty)$.

This is true because:

$$ 1+\log(u)=1+\int_1^u\frac1tdt\le1+\int_1^u1dt=1+u-1=u $$

Some care is needed to establish that the inequality is true for both $u\ge1$ and $0<u\le1$. In the second case, we can see this more clearly by writing:

$$ 1+\int_1^u\frac1tdt=1+\int_u^1-\frac1tdt\le1+\int_u^1-1dt=1-1+u=u $$

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Completing glebovg's answer :

  • the inequality $1+x \le e^x$ clearly holds for $x \leq -1$,

  • suppose $x \geq -1$ :

the series $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ can be written (grouping the terms in pair) :

$$e^x = 1 + x + \sum_{k \geq 1} \left( \frac{x^{2k}}{(2k)!} + \frac{x^{2k+1}}{(2k+1)!} \right)$$

$$e^x = 1 + x + \sum_{k \geq 1} x^{2k}\left( \frac{1}{(2k)!} + \frac{x}{(2k+1)!} \right)$$

$$e^x = 1 + x + \sum_{k \geq 1} x^{2k}\left( \frac{2k + 1 + x}{(2k+1)!} \right)$$

under the assumption $x \geq -1$, the $\sum$ part is clearly a sum of positive numbers.

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For positive values ​​of $ x $ We can use the following characterization of $e^x$ $$ e^x=\lim_{t\to \infty} \Big( 1+\frac{1}{t}\Big)^{tx},\quad t> 0,x\geq 0. $$ The Bernoulli's inequality states that $(1 + y)^r \geq 1 + ry$ for every $r \geq 0$ and every real number $y \geq −1$. Then for $y=\frac{1}{t}$ and $r=tx\geq 0$, \begin{align} e^x= &\lim_{t\to \infty} \Big( 1+\frac{1}{t}\Big)^{tx}\\ \geq &\lim_{t\to \infty} \Big(1+\frac{1}{t}(tx) \Big)\\ = & 1+x \end{align}

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The series expansion of $(1+x)$ is $(1+x)$, while $\exp(x)=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+...$. Subtracting the second from the first one you have the difference $d=\exp(x)-(1+x)=\frac{1}{2}x^2+\frac{1}{6}x^3+...$ which is zero only for $x=0$ otherwise $d\gt 0$ Q.E.D.

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Same comment, is it trivially evident for negative $x$ that $d>0$? –  Macavity Sep 25 '13 at 10:17
    
@Macavity: make the substitution: $x\to -y$. Anyway it's not trivial. –  Riccardo.Alestra Sep 25 '13 at 10:26
    
I have seen that proof done considering three regions separately - viz. $x \ge 0, -1 < x < 0, x \le -1$. The first region is trivial, the rest two not so much :( –  Macavity Sep 25 '13 at 10:34

Let $f(x)=\exp(x)-x-1$. Then, $f'(0)=0$. But $f$ is strictly convex (a difference of a strictly convex function and an affine one), so that $0$ most be a unique global minimum. Hence, $\exp(x)-x-1=f(x)\geq f(0)=0$ for all $x\in\mathbb{R}$.

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We want to show that (1) $$1+x\leq e^x,$$ for $x\in\mathbb{R}$. When $x\geq 0$, we have $$1+x\leq 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=e^x.$$ Suppose $x=-X$, where $X>1$, then $1+x=1-X<0$ and $e^{x}=e^{-X}=1/e^X>0$. Hence (1) holds.

Now take logarithms of (1) to obtain $$\log(1+x)\leq x.$$ But $$\log(1+x) = x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots,$$ where $|x|<1$. Suppose $x=-X$, where $0<X<1$, then $$\log(1+x)=\log(1-X)=-X-\frac{X^2}{2}-\frac{X^3}{3}-\cdots<-X=x.$$ Or, equivalently, $$1+x<e^x,$$ where $-1<x<0$.

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Third line, I think you meant "Suppose $x=-X$" and not "Suppose $x<-X$". –  user37238 Sep 25 '13 at 12:49

One which uses $\exp(x) = \frac 1{\exp(-x)} $ $$ 1 + x \underset{ \text{obvious}\\ \text{for $x>0$}}{\lt} 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + ... = {1 \over 1 - x + {x^2 \over 2!} - { x^31 \over 3!} + ... }$$ Now we replace $+x$ by its negative counterparts and get similarily $$ 1 - x \underset{ \quad \text{for $x>0$}\\ \text{but not obvious}}{\lt} 1 - x + {x^2 \over 2!} - {x^3 \over 3!} + ... = {1 \over 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + ... }$$ But now the comparision with the fraction on the rhs becomes obvious if we look at the reciprocals. The reciprocal ${1\over 1-x}=1+x+x^2+x^3+...$ is and we get $$ {1 \over 1 - x} = 1+x+x^2+... \underset{ \text{obvious}\\ \text{for $x>0$}}{\gt} 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + ... = {1 \over 1 - x + {x^2 \over 2!} - {x^3 \over 3!} + ... }\\ $$

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upps, I see there was another answer earlier, but which seemed too short for me to not only skim over it... –  Gottfried Helms Sep 26 '13 at 9:28

$1+x \le e^x$

Take the ln of both sides

$ln(1+x) \le x$

differentiate both side w.r.t $x$

$\frac{1}{1+x} \le 1$

which holds $\forall x \in \mathbb{R}, x \neq -1$.

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You should work this proof in the opposite direction for it to be an actual proof. –  Cameron Williams Sep 25 '13 at 14:51
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$e^{-x}<0$. Now differentiate both sides w.r.t. $x$ to get $-e^{-x}<0$, which holds $\forall x\in\mathbb R$. So our original statement is correct, is it? –  Donkey_2009 Sep 25 '13 at 14:54
    
@donkey: How can $e^{-x} \lt 0$ ? –  Gottfried Helms Sep 26 '13 at 8:27
    
@GottfriedHelms - I just proved it! Just like the OP, I differentiated both sides with respect to $x$ and arrived at a manifestly true statement. So it must be true! –  Donkey_2009 Sep 26 '13 at 9:05
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@donkey: put any real number $x$ into your calculator and compute $\exp(x)$. The result will always be greater than zero, irrespectively of whether $x$ is positive or negative. So there must be a flaw in your proof... –  Gottfried Helms Sep 26 '13 at 9:19

Proof by induction (works for natural numbers)

Assume it works for n

1 + n < e^n

Then we prove that it works for n+1

1 + (n+1) < e^(n+1)  

Proof

         1 + n < e^n 
  or    1 + n + 1 < e^n + 1 
  or    1 + n + 1 < e^n + e^n   since e^n > 1
  or    1 + n + 1 < e^n * 2
  or    1 + (n+1) < e^n * e       since e > 2
  or    1 + (n+1) < e^(n+1)

hence it is true for n+1 if true for n. We know it is true for 1, hence by induction is true of 2, 3, 4...so on.

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This isn't really a proof. Like, where is your inductive step? –  Alexander Sep 25 '13 at 23:00
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updated the answer works for natural number only though –  tihom Sep 25 '13 at 23:20

The fact $\frac{d}{dx} e^x = e^x$ is nicely demonstrated using the self-similar nature of exponential functions. (See my answer here.)

This justifies (actually, declares) that $y=x+1$ is tangent to $y=e^x$; thereafter, since the slope increases (or decreases) as $x$ gets larger (respectively, smaller) the line and curve cannot meet again (which is an informal way of stating the convexity property).

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Proof by contradiction: Let us assume that $ 1+x > e^x (x >= 0 ; x <= -1)$

Now, taking log both sides, $log(1+x) > x$ . Now differentiating both sides,

$ 1/(1+x) > 1$

which is impossible for x belonging to the set mentioned above. Hence, $1+x>= e^x$; for $x>=0; x<=-1 $

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This is wrong as written. Differentiation does not preserve inequalities –  leo Sep 26 '13 at 18:01

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