Need help with three proofs.

Question

Proof#1:
Prove that |x+y| is less than or equal to |x|+|y|

Proof#2:
Prove that |x|-|y| is less than or equal to |x-y|

Proof#3:
Prove that |x+y+z| is less than or equal to |x|+|y|+|z|.

Answer 1

|x+y|
two possibilities
A:
x+y
B:
-x-y = -(x+y)

if A:
x+y is maximum when both are +
then x+y = |x|+|y|

if B:
-(x+y) has same absolute value as A

etc

Answer 2

google "triangle inequality" for many discussions and proofs