Prove the linear independence of $\{v_1,v_2\}\,$?

Prove that if $\{v_1,v_2\}$ are linearly independent (LI), then also $\{v_1,v_1+v_2\}$ are linearly independent.
Is the reverse true?

I have proved the first part, but now i am stuck on whether the reverse is true.

so if the $ \{ v_1,v_1+v_2 \} $ are LI, are also $\{v_1,v_2\}$ LI? How can i prove this?

Thanks in advance.

The reverse can be proved with the usual definition of linear independence.

Let $c_1,c_2\in\mathbb{R}$ be such that $c_1 v_1+c_2v_2=0$. Then $$0=c_1v_1+c_2v_2=c_1v_1+c_2v_2+c_2v_1-c_2v_1=(c_1-c_2)v_1+c_2(v_1+v_2)$$

Because $v_1,v_1+v_2$ are linearly independent, we must conclude $c_1-c_2=c_2=0$, which leads to $c_1=c_2=0$ and proves $v_1,v_2$ are linearly independent.

$\{v_1 ,v_1 +v_2 \}$ L.I $\implies \{v_1,v_2 \}$ L.I

Proof : Assume the contrary that $\{v_1,v_2 \}$ L.D

Then, $v_2 =\lambda v_1$ for some scalar $\lambda$

Then $v_1+ v_2 = (1+\lambda ) v_1$

And this implies $\{v_1 ,v_1 +v_2 \}$ is linearly dependent.