Is there some article or chapter in some book which covers proving all usual properties of exponential function with definition of real exponentiation via showing that sequence $\{a^{x_n}\}$ converges when sequence $\{x_n\}$ of rationals converges and the value of the limit of $\{a^{x_n}\}$ is independent of the sequence $\{x_n\}$, where $a>0$ ?
Thanks in advance.
2026-03-29 15:14:46.1774797286
Definition of real exponent
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This is in Terence Tao's book "Analysis I". See section 6.7, p152, third edition. Lemma 6.7.1 captures the essence of your question.