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Math Help - Mathematical Induction

  1. #1
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    Mathematical Induction

    Suppose that x1, x2, x3, ... ,xn are real numbers. Prove (using mathematical induction) that
    l x1 + x2 + ... + xn l < lx1l + lx2l + ... + lxnl
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    Quote Originally Posted by bearej50 View Post
    Suppose that x1, x2, x3, ... ,xn are real numbers. Prove (using mathematical induction) that
    l x1 + x2 + ... + xn l < lx1l + lx2l + ... + lxnl
    You first need to show |x+y|\leq |x|+|y|
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    I know this to be true. I can use this theorem in my proof. I just dont know how to write a proof for this using mathematical induction.
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    Quote Originally Posted by bearej50 View Post

    I know this to be true. I can use this theorem in my proof. I just dont know how to write a proof for this using mathematical induction.
    You will prove this for n\geq 2. If this statement is true for k variables i.e. |x_1+...+x_k| \leq |x_1| + ... + |x_k| we shall prove it is true for k+1 variables. In the expression |x_1+...+x_k+x_{k+1}| think of it as |(x_1+...+x_k)+x_{k+1}| but this is less than or equal to |x_1+...+x_k| + |x_{k+1}| but this is less than or equal to |x_1|+...+|x_k|+|x_{k+1}|.
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    thank you again
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