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log sequence converges, limit is continuous

Source: Putnam 1982 B5

October 7, 2021
functionlimitsreal analysisSequencesanalysis

Problem Statement

For each x>eex>e^e define a sequence Sx=u0,u1,…S_x=u_0,u_1,\ldots recursively as follows: u0=eu_0=e, and for n≥0n\ge0, un+1=log⁡unxu_{n+1}=\log_{u_n}x. Prove that SxS_x converges to a number g(x)g(x) and that the function gg defined in this way is continuous for x>eex>e^e.
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