I am trying to understand the answer to a question. The question is:
I am given the following set [MATH]A= \lbrace x>0 : x^2>4 \rbrace = \lbrace x>0 : x>2 \rbrace[/MATH] and told to determine,
a) lower bound
b) Let [MATH]L[/MATH] be a lower bound s.t. [MATH]L>2[/MATH] and let [MATH]y=\frac{L+2}{2}[/MATH]. Show that [MATH]L>y>2[/MATH]c) Show that [MATH]y \in A[/MATH] and [MATH]L \leq y[/MATH]. Show this leads to contradition, in which case [MATH]L \leq 2 \Rightarrow \inf{A}=2[/MATH]
So the answer given for a) is that 2 is the lower bound, no issues with that
The answer for b) though I do not understand part of it, which is:
Since [MATH]L>2 \Rightarrow L+2>4 \text{, in which case } y=\frac{L+2}{2} >2[/MATH]. Also [MATH]y=\frac{L+2}{2} > \frac{L+L}{2}=L [/MATH]
Firstly, I do not understand "[MATH]L>2 \Rightarrow L+2>4[/MATH]" and secondly I do not understand "[MATH]y=\frac{L+2}{2} > \frac{L+L}{2}=L [/MATH]"
Could somebody expand the answer for me perhaps?
I am given the following set [MATH]A= \lbrace x>0 : x^2>4 \rbrace = \lbrace x>0 : x>2 \rbrace[/MATH] and told to determine,
a) lower bound
b) Let [MATH]L[/MATH] be a lower bound s.t. [MATH]L>2[/MATH] and let [MATH]y=\frac{L+2}{2}[/MATH]. Show that [MATH]L>y>2[/MATH]c) Show that [MATH]y \in A[/MATH] and [MATH]L \leq y[/MATH]. Show this leads to contradition, in which case [MATH]L \leq 2 \Rightarrow \inf{A}=2[/MATH]
So the answer given for a) is that 2 is the lower bound, no issues with that
The answer for b) though I do not understand part of it, which is:
Since [MATH]L>2 \Rightarrow L+2>4 \text{, in which case } y=\frac{L+2}{2} >2[/MATH]. Also [MATH]y=\frac{L+2}{2} > \frac{L+L}{2}=L [/MATH]
Firstly, I do not understand "[MATH]L>2 \Rightarrow L+2>4[/MATH]" and secondly I do not understand "[MATH]y=\frac{L+2}{2} > \frac{L+L}{2}=L [/MATH]"
Could somebody expand the answer for me perhaps?