Can the product of infinitely many elements from Q\mathbb Q be irrational?

I know there are infinite sums of rational values, which are irrational (for example the Basel Problem). But I was wondering, whether the product of infinitely many rational numbers can be irrational.
Thank you for your answers.

Answer

Yes, it can.

Consider any sequence (a_n) of non-zero rational numbers which converges to an irrational number. Then define the sequence b_n by b_1 = a_1 and

b_n = \frac{a_n}{a_{n-1}}

for n > 1.

We then have that

b_1 b_2 \cdots b_n = a_1 \frac{a_2}{a_1} \frac{a_3}{a_2} \cdots \frac{a_n}{a_{n-1}} = a_n.

We thus see that every term of (b_n) is rational, and that the product of the terms of (b_n) is the same as the limit of a_n, which is irrational.

Attribution
Source : Link , Question Author : Mister Set , Answer Author : Dylan

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