# Mathematical Induction

• Feb 10th 2009, 09:24 AM
bearej50
Mathematical Induction
Suppose that x1, x2, x3, ... ,xn are real numbers. Prove (using mathematical induction) that
l x1 + x2 + ... + xn l < lx1l + lx2l + ... + lxnl
• Feb 10th 2009, 04:41 PM
ThePerfectHacker
Quote:

Originally Posted by bearej50
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 $\displaystyle |x+y|\leq |x|+|y|$
• Feb 16th 2009, 10:12 AM
bearej50
http://www.mathhelpforum.com/math-he...728e1577-1.gif
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.
• Feb 16th 2009, 10:26 AM
ThePerfectHacker
Quote:

Originally Posted by bearej50
http://www.mathhelpforum.com/math-he...728e1577-1.gif
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 $\displaystyle n\geq 2$. If this statement is true for $\displaystyle k$ variables i.e. $\displaystyle |x_1+...+x_k| \leq |x_1| + ... + |x_k|$ we shall prove it is true for $\displaystyle k+1$ variables. In the expression $\displaystyle |x_1+...+x_k+x_{k+1}|$ think of it as $\displaystyle |(x_1+...+x_k)+x_{k+1}|$ but this is less than or equal to $\displaystyle |x_1+...+x_k| + |x_{k+1}|$ but this is less than or equal to $\displaystyle |x_1|+...+|x_k|+|x_{k+1}|$.
• Feb 17th 2009, 11:30 AM
bearej50
thank you again