Suppose $a,b$ and $c$ are linearly independent vectors in a vector space $V$. How can I prove that $a+b$ or $b+c$ are also linearly independent?
If I understand your question correctly, you want to show that if $a$, $b$, $c$ are linearly independent, then $a+b$ and $b+c$ are linearly independent.
Just look at the definitions.
You know that $x_1a+x_2b+x_3c=0$ implies $x_1=x_2=x_3=0$. (This is the definition of linear independence for three vectors.)
You ask whether $y_1(a+b)+y_2(b+c)=0$ implies $y_1=y_2=0$.
Just simplify this to get: $y_1 a + (y_1+y_2)b +y_2c=0$. This implies that $y_1=y_1+y_2=y_2=0$. The condition $y_1+y_2=0$ is redundant there, but we have shown that $y_1=y_2=0$.
This means that the vectors $a+b$, $b+c$ are linearly independent.