Given that \( \{a_n\}\) is a sequence of real numbers such that the limit, \( \displaystyle \lim_{n\to\infty} \dfrac{\ln a_n}{2^n} \) exists.

Prove that the following infinitely nested functions also converges.

\[ \displaystyle \sqrt{a_{1} + \sqrt{a_{2} + \ldots + \sqrt{a_{n}}}}\]

This is a part of the set Formidable Series and Integrals

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