Let $\{x_n\}$ be a sequence. Suppose that there are two convergent subsequences $\{x_{n_i}\}$ and $\{x_{m_i}\}$. Suppose that $\lim_{i\to\infty} x_{n_i} =a$ and $\lim_{i\to\infty} x_{m_i} =b$ where $a \neq b$. Prove that $\{x_n\}$ is not convergent.
I cannot use the fact that if a sequence is convergent, then its subsequences are convergent and the limit of the sequence is equal to the limit of the subsequence. I am stuck figuring out how to prove that it is not convergent a different way.