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This MathOverflow post cites that Legendre allegedly showed that given $a_{i}\in\mathbb{Z}\setminus\left\{0\right\}, b_{i}\in\mathbb{Z}$,

$$\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}}$$

is irrational if $|a_i|<|b_i|$, for any sufficiently large $i$. Can anyone prove this or provide a reference?

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The reference to Legendre's proof is: A. M. Legendre, Essai sur la théorie des nombres, (1798). For a modern discussion, see On a Theorem of Legendre in the theory of continued fractions (1994).

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