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Thread: Number System Proofs

  1. #1
    Jan 2009

    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:

    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
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  2. #2
    o_O is offline
    Primero Espada
    o_O's Avatar
    Mar 2008
    I'm going to use this: $\displaystyle x < y \ \Rightarrow \ x + z < y + z \qquad \forall x,y,z \in \mathbb{R}$

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

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