Induction Proof: l x1 + x2 + ... + xn l [u]< [/u]lx1l +....

[bearej50]

Suppose that x1, x2, x3, ... ,xn are real numbers. Prove (using mathematical induction) that

l x1 + x2 + ... + xn l < lx1l + lx2l + ... + lxnl

 

[daon]

Show it for n=1,2. This may take a few cases and on how you defined absolute value.

IH: |x1+...+xn| < |x1| + ... + |xn| for n<=k.

Let Y = x1+...+xk, and note by our IH, |Y| <= |x1|+...+|xk|. (Y is just a real number now, expressable as this finite sum)

|x1 + ... + xk + x(k+1)| = |Y+x(k+1)| <= |Y| + |x(k+1)| <= |x1|+|x2|+...+|xk| + |x(k+1)|