Let $\{x_n\}$ be a sequence. Suppose that there are two convergent subsequences $\{x_{n_{\Large{i}}} \}$ and $\{x_{m_{\Large{i}}} \}$. Suppose that $\lim\limits_{i\to\infty} x_{n_{\Large{i}}} = a$ and $\lim\limits_{i\to\infty} x_{m_{\Large{i}}} = b$, where $a \ne b$. Prove that $\{x_n\}$ is not convergent.
I have to prove this without using the idea that subsequences are convergent if the sequence is convergent and their limits are equal. I have no idea where to go with this problem and have been stuck on it for quite some time now.