Thursday, 8 October 2015

real analysis - Prove that if f is eventually monotone and eventually bounded Rightarrowlimxrightarrowinftyf(x) is finite

If the function f is defined on an unbounded above domain D 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

No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...