Consider the following statements:

1.) If $\displaystyle f : A \rightarrow \mathbb{R}$ is uniformly continuous, then $\displaystyle f(A)$ is bounded.

2.) If $\displaystyle f : A \rightarrow \mathbb{R}$ is uniformly continuous, and also if $\displaystyle A$ is bounded, then $\displaystyle f(A)$ is bounded.

Either prove, or disprove, the statements above.