Real Analysis: Prove BV function bounded and integrable

**1. The problem statement, all variables and given/known data**

f is of bounded variation on [a;b] if there exist a number K such that

|f(a_{k})-f(a_{k-1})| K

a=a_0<a_1<...<a_n=b; the smallest K is the total variation of f

I need to prove that

1) if f is of bounded variation on [a;b] then it is bounded on [a;b]

2) if f is of bounded variation on [a;b], then it is integrable on [a;b]

**2. The attempt at a solution**

i thought of using triangle inequation such that

0<=|f(b)-f(a)|<= |f(a_{k})-f(a_{k-1})| K

but im not really sure how to prove the two statements

any help is really appreciated

thanks