I have attempted to prove the following Theorem is my argument correct?
NOTE: The argument below makes use of the following notation and theorems.
$T\in\mathcal{L}(V)$ denotes that $T$ belongs to the set of all linear operators on the vector space $V$.
$5.10$ Given that $T\in\mathcal{L}(V)$ such that $\lambda_1,\lambda_2,...,\lambda_m$ are distinct eigenvalues of $T$ then the corresponding list of eigenvectors $v_1,v_2,...,v_m$ is linearly independent.
Theorem. Given that $T$ is a linear operator over the vector space $V$ where $u$ and $v$ are eigenvectors of $T$ such that $u+v$ is also an eigenvector of $T$, then $u$ and $v$ are eigenvectors corresponding to the same eigenvalue.
Proof. Assume on the contrary that $u$ and $v$ are eigenvectors of $T\in\mathcal{L}(V)$ such that they do not correspond to the same eigenvalue consequently $$Tu = \lambda_1u\ \operatorname{and}\ Tv = \lambda_2v\ \operatorname{where}\ \lambda_1\neq\lambda_2$$
more over since $u+v$ is an eigenvector of $T$ it follows that $T(u+v) = \lambda_3(u+v)$. The Linearity of $T$ implies that $T(u+v) = Tu+Tv = \lambda_1u+\lambda_2v = \lambda_3(u+v)$ which implies that $$(\lambda_3-\lambda_1)u+(\lambda_3-\lambda_2)v = 0$$
but $\lambda_1\neq\lambda_2$ which by $5.10$ implies that $u$ and $v$ are linearly independent consequently $$\lambda_3-\lambda_1 = 0$$ $$\lambda_3-\lambda_2 = 0$$ which implies that $\lambda_1 = \lambda_2$ a contradiction.
$\blacksquare$
Source: Linear Algebra Done Right by Sheldon Axler 5-A Exercise 25.
It is perfect, except that if you look closely there is no need to work by contradiction. You never actually use the fact that $\lambda_1\neq\lambda_2$, therefore you can remove this assumption and simply come to the conclusion that $\lambda_1= \lambda_2$ without changing a word.
Edit since it was not clear for everyone: "if the vectors are linearly dependent then obviously $\lambda_1=\lambda_2$, and if they are not (insert your own words here) again $\lambda_1=\lambda_2$".