Problem 273 (difficulty: 5/10)

Prove that if

(i) \(\displaystyle a_n\to a\ge1\) and \(\displaystyle (b_n)\) is bounded, then

\(\displaystyle \Olim a_n^{b_n} = a^{\Olim b_n} \quad\text{and}\quad \Ulim a_n^{b_n} = a^{\Ulim b_n}. \)

(ii) \(\displaystyle a_n\to a\le 1\) and \(\displaystyle (b_n)\) is bounded, then

\(\displaystyle \Olim a_n^{b_n} = a^{\Ulim b_n} \quad\text{and}\quad \Ulim a_n^{b_n} = a^{\Olim b_n}. \)

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