If the function f is defined on an unbounded above domain D⊆ℜ and is eventually monotone and eventually bounded, then lim is finite
I tried to workout the proof as:
Since f is eventually monotone \Rightarrow \exists x^*, x^* \leq x_1 < x_2 we have f(x_1) \leq f(x_2)
and since f is eventually bounded \Rightarrow \exists \hat{x},\ \exists \ L \leq M \in \mathbb{R} \ s.t. \lim_{x\rightarrow \infty} f(x) = L \\\forall \ \hat{x}\leq x
Take x = max(\hat{x}, x^*) and we have \lim_{x\rightarrow \infty} f(x) = L
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