Would this be a legitimate complex proof?? (1 Viewer)

[Michaelmoo]

Hey. Ok. I've done a proof but I'm not sure if it would be accepted in an exam. I mean it uses logic to prove a particular solution.

Show that |z1 +z2 +..................+zn| < |z1| + |z2| + ............+ |zn|

My proof:

It is known that
|z1 + z2| < |z1| + |z2| ----- (1)
|z3 + z4| < |z3| + |z4| ------(2)
.......
|zn-1 + zn| < |zn-1| + |zn| if n is even or |zn| = |zn| is n is odd ------(k)

1 + 2 + ...........+ k

|z1 + z2| +|z3 + z4| +..........+ |zn-1 + zn| < |z1| + |z2| + |z3| + |z4| +...+|zn|

now from this:
|z1 + z2 + z3 +z4| < |z1 + z2| +|z3 + z4| + < |z1| + |z2| + |z3| + |z4|

Continue the series and therefore:

|z1 +z2 +..................+zn| < |z1| + |z2| + ............+ |zn|

I understand that the best way to do this is by MI. But we havn't done MI nor sequences and series. If there is any other way you could do it, that would be great.

Thanks.

 

[Trebla]

Looks fine I think. That method pretty much follows the principle of mathematical induction without explicity setting out an induction proof lol.
A similar way to your approach is:
|z₁ + [z₂ + z₃ + ....... + z_n]| ≤ |z₁| + |z₂ + z₃ + ....... + z_n|
But:
|z₂ + z₃ + ....... + z_n| ≤ |z₂| + |z₃ + z₄ + ....... + z_n|
≤ |z₂| + |z₃| + |z₄ + z₅ + ....... + z_n|
.
.
.
≤ |z₂| + |z₃| + |z₄| + |z₅| + ....... + |z_n|

.: |z₁ + [z₂ + z₃ + ....... + z_n]| ≤ |z₁| + |z₂ + z₃ + ....... + z_n| ≤ |z₁| + |z₂| + |z₃| + |z₄| + |z₅| + ....... + |z_n|

 

[Michaelmoo]

Looks fine I think. That method pretty much follows the principle of mathematical induction without explicity setting out an induction proof lol.
A similar way to your approach is:
|z₁ + [z₂ + z₃ + ....... + z_n]| ≤ |z₁| + |z₂ + z₃ + ....... + z_n|
But:
|z₂ + z₃ + ....... + z_n| ≤ |z₂| + |z₃ + z₄ + ....... + z_n|
≤ |z₂| + |z₃| + |z₄ + z₅ + ....... + z_n|
.
.
.
≤ |z₂| + |z₃| + |z₄| + |z₅| + ....... + |z_n|

.: |z₁ + [z₂ + z₃ + ....... + z_n]| ≤ |z₁| + |z₂ + z₃ + ....... + z_n| ≤ |z₁| + |z₂| + |z₃| + |z₄| + |z₅| + ....... + |z_n|

Ahh yes. Your method is much more logical and easier to comprehend, rather than having simultaneous equations etc. Just wanted to make sure that it should be ok.

Thanks.

 

[Michaelmoo]

Anyone else think this would be considered correct? I'm still a little unsure if it would be accepted. I don't know if you could just make a substitution like that.