Let
![a_{1}, \ldots, a_{n}](/media/m/a/3/9/a39b3d0e16f2d294f2f3c36ad6fe328c.png)
be an infinite sequence of strictly positive integers, so that
![a_{k} < a_{k+1}](/media/m/e/a/4/ea48898b97858ee367b6a2e1c836510e.png)
for any
![k.](/media/m/3/5/d/35d6ded5ad555e4f371e82635560eb35.png)
Prove that there exists an infinity of terms
![a_m](/media/m/4/c/9/4c9d7d7976c2e05720cb3a3b7e878a00.png)
, which can be written like
![a_m = x \cdot a_p + y \cdot a_q](/media/m/f/c/3/fc38c5029c78c345d259a28d0669fe8f.png)
with
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
strictly positive integers and
![p \neq q](/media/m/2/f/d/2fd4d3de287059b8a1e9684675366ae8.png)
.
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Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $a_m$, which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q$.