<s>[INST] Applications of the inequality $e^x\geqslant 1+x$ I am interested in proofs of famous theorems or inequalities which can be proved by the inequality
$$
e^x\geqslant 1+x.
$$  
For example, the divergence of harmonic series can be proved by assigning $x=\frac{1}{k}$.
$e^{\pi}>\pi^e$ can be proved by assigning $x=\frac{\pi}{e}-1$.  
The inequality of the arithmetic average of n-th degree can be proved by assigning $x=\frac{a_k}{(a_1*a_2*\cdots*a_n)^{\frac{1}{n}}} -1$.
The divergence of $\frac{e^t}{t^n}$ can be proved by assigning $x=\frac{t}{n+1}$.  

I want to know other possible proofs.
 [/INST] Define $p_i$ as the i-th prime number. Taking $x=\frac{1}{p_i -1}$ and we have 
$$e^{\frac{1}{p_i-1}}≧1+\frac{1}{p_i -1}=1+\frac{1}{p_i} +\frac{1}{p_i ^{2}}+\cdots.$$
The product from $i=1$ to $n$ is
$$e^{1+\sum_{i=2}^n \frac{1}{p_i-1}}≧\prod_{i=1}^n \left(1+ \frac{1}{p_i} +\frac{1}{p_i ^{2}}+\cdots \right).$$
And here,
$$\prod_{i=1}^n \left(1+ \frac{1}{p_i} +\frac{1}{p_i ^2}+\cdots \right)≧\sum_{k=1}^{x(n)} \frac{1}{k},$$
$$\sum_{i=2}^n \frac{1}{p_i-1}≦ \sum_{i=1}^{n-1} \frac{1}{p_i}.$$  
So, we can say
$$e^{\sum_{i=1}^{n-1} \frac{1}{p_i}}≧ \sum_{k=1}^{x(n)} \frac{1}{k},$$
$$\Leftrightarrow\sum_{i=1}^{n-1} \frac{1}{p_i}≧\log \sum_{k=1}^{x(n)} \frac{1}{k}.$$ 
When $n\to\infty$, $x(n)\to\infty$ and $\lim_{n\to\infty} \log \sum_{k=1}^{x(n)} \frac{1}{k}=\infty$. So,
$$\sum_p \frac{1}{p}=\infty.$$ 
That is how we can prove the sum of prime numbers diverges.
</s>