6

Part of ICMC 6

Problems(1)

Exponential sequence converges in fractional parts

Source: ICMC 2023 Round 1 P6

Consider the sequence defined by

a_1 = 2022

a_{n+1} = a_n + e^{-a_n}

n \geq 1

. Prove that there exists a positive real number

r

for which the sequence

\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . .

converges.
Note:

\{x \} = x - \lfloor x \rfloor

denotes the part of

x

after the decimal point.
Proposed by Ethan Tan

ICMCcollege contestsreal analysisConvergenceSequencesfloor function