Triangle inequality

 Theorem:-     (Triangle inequality)
for all a,b,in R,Then|a+b|< |a|+|b|.

Proof :-

We have, |a|=a if a>0 and |a|= -a if a<0 now taking |a|=c

and we get required result

 -|a| ≤ a ≤ |a|....................... (1)

|-b| ≤ b ≤   |b|……………..  (2)

adding equation (1) and (2)

We get

-(|a| + |b|) ≤  |a + b| ≤ |a| +|b|

Therfor

  |a+b | ≤  |a | + |b |