Results 1 to 2 of 2

Math Help - induction proof

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    12

    induction proof

    How would you use induction to show that for  a_{i}\epsilon R

    ||a_{1} + a_{2} + ... + a_{n}|| \leq ||a_{1}|| + ||a_{2}|| + ... + ||a_{n}|| ?

    This seems like a very general induction proof. Thank you. Barton.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Barton View Post
    How would you use induction to show that for  a_{i}\epsilon R

    ||a_{1} + a_{2} + ... + a_{n}|| \leq ||a_{1}|| + ||a_{2}|| + ... + ||a_{n}|| ?

    This seems like a very general induction proof. Thank you. Barton.
    First poove as a base case that for all a_1, a_2 \in \mathbb{R} :

    ||a_1+a_2||\le ||a_1||+||a_2||

    Now suppose that for some k and for all a_1, .. a_k \in \mathbb{R} that:

    ||a_1+ .. +a_k||\le ||a_1||+ .. + ||a_k||.

    Now consider:

    ||a_1+ .. +a_k+a_{k+1}||=||(a_1+ .. + a_k)+a_{k+1}|| \le ||a_1+ .. + a_k||+||a_{k+1}||

    by the base case, and so:

    ||a_1+ .. +a_k+a_{k+1}||=||(a_1+ .. + a_k)+a_{k+1}|| \le ||a_1+ .. + a_k||+||a_{k+1}|| \ \le ||a_1|| + .. + ||a_k||+||a_{k+1}||

    by the assumption.

    You should now be able to put this all together to give the induction proof.

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof by Induction
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: October 11th 2011, 07:22 AM
  2. Proof by Induction
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 16th 2010, 12:09 PM
  3. Mathemtical Induction Proof (Stuck on induction)
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: March 8th 2009, 09:33 PM
  4. Proof by Induction??
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 6th 2008, 03:55 PM
  5. Proof with algebra, and proof by induction (problems)
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: June 8th 2008, 01:20 PM

Search Tags


/mathhelpforum @mathhelpforum