Solving Induction Proof w/ Homework: Step by Step Guide

[ar6]

Homework Statement

http://img87.imageshack.us/img87/469/17317851uu7.jpg

Homework Equations

The Attempt at a Solution

First I have to show that P(1) is true
|a1| ≤ |a1| This is true

Assume above is true for all n.

Show that

|∑i=1n+1 ai| ≤ n+1∑i=1 |ai|


|a1 + a2 + a3…+ an + an+1| ≤ |a1| + |a2| + |a3| +…+ |an| + |an+1|

|∑i=1n ai + an+1| ≤ n+1∑i=1 |ai| + |an+1|

assuming I'm on the right track. How can i proceed. This problem is so abstract its driving me crazy. Sorry for formatting.

 

[MalayInd]

Homework Statement

http://img87.imageshack.us/img87/469/17317851uu7.jpg

Homework Equations

The Attempt at a Solution

First I have to show that P(1) is true
|a1| ≤ |a1| This is true

Assume above is true for all n.

Show that

|∑i=1n+1 ai| ≤ n+1∑i=1 |ai|


|a1 + a2 + a3…+ an + an+1| ≤ |a1| + |a2| + |a3| +…+ |an| + |an+1|

.

you have to arrive at the last equation using the preceding equation.
Lets start with the left hand side of last equation.
|a1 + a2 + a3…+ an + an+1|≤|a1 + a2 + a3…+ an |+ |an+1|
.......{ since it is valid for n=2}
|a1 + a2 + a3…+ an + an+1|≤|a1| + |a2 |+ |a3|+…+ |an |+ |an+1|
........{since valid for n=n(seond last equation i your post)}

Keep Smiling
Malay

 

[ZioX]

You use a strong induction. Assume true for all k<n. Then

\left|\sum_i a_i\right|=\left|\sum_{i=1}^{n-1}a_i +a_n\right|

You should see where to go from here.Problems like these are often trivial inductions. Like, for example, showing that taking the intersection of two open sets is open immediately implies that the finite intersection of open sets is always open. Why? Well, for n=3 we have (A n B)nC but we showed AnB is open! So we just repeatedly use this fact.

It's the same thing here, really.

Edit: If you know nothing about open sets, completely disregard.

 

[learningphysics]

You should demonstrate that it is true for n = 2...

MalayInd showed the main inductive step which is valid for n+1>2 but not for n+1 = 2.

 

[MalayInd]

for n+1=2, it reduces to triangle inequality.