There are millions of years of fossils showing that death was around long before Adam and Eve. In any case, it is impossible to have life without death. Anything eating plants would be causing the death of tiny animals necessary for plant life, not to mention the death of plants themselves. Then there are the soil nutrients that plants need which are formed by the death of other organisms. So there was death in the world before Adam and Eve.
Sometimes people think Romans 5:12 says otherwise:
Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men because all sinned—
A similar idea is in 1 Corinthians 15:21. These verses are referring to the introduction of sin into the world, and the punishment of sin which is death (Rom 6:23). From the time of Adam and Eve, people have sinned and suffered the consequences by dying. However personal sin is not the cause of all death. Just think about Jesus who was killed, or young babies who can still die even though they have never sinned. Instead, Paul is referring to the death that is a result of sin. This death was introduced at the time of Adam and Eve when the first sin occurred.