Uniqueness of Limits and Boundedness of Convergent Real Sequences

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of real numbers. Then the following hold.

1. (Uniqueness of limits) If $(a_{n})$ converges to $A$ and also converges to $A'$, then $A=A'$.

2. (Convergent sequences are bounded) If $(a_{n})$ converges to some real number, then $(a_{n})$ is bounded.