1. ## Number System Proofs

Hi guys, I sometimes troll this forum for information but never post. But I'm really stumped in this 2nd year analysis course. When it comes to proving stuff I just completely suck! I've taken Calc 1-4; Differential Eqns; Linear Algebra; and this is by far the hardest course because I can't seem to wrap my head around the ideas of proving this stuff! Anyway, here are two questions that I hope someone can answer, I'm pretty sure once I see how to do them, I'll be ok.

Question 1:
Let a and b be any real numbers such that a<b and let t be their arithmetic mean. Prove that a<t<b. (t=1/2(a+b))

Question 2:
Solve the inequalities. Justify each step:
x≤5x-4<2x

Now I'm pretty sure these aren't hard questions, but I just don't understand how you can show that.

Any help would be appreciated. Thanks

2. I'm going to use this: $x < y \ \Rightarrow \ x + z < y + z \qquad \forall x,y,z \in \mathbb{R}$

So, adding $a$ to both sides: $a < b \ \Rightarrow \ a + a < a + b \ \Leftrightarrow \ 2a < a + b$

But also, adding $b$ to both sides: $a < b \ \Rightarrow \ a + b < b + b \ \Leftrightarrow \ a + b < 2b$