Genesis 9:7

"'As for you, be fruitful and increase in number; multiply on the earth and increase upon it.'" NIV translation

Once again, God's instruction for Noah and his family is for them to repopulate the earth.

Genesis 9:8 - 11

"Then God said to Noah and to his sons with him: 'I now establish my covenant with you and with your descendants after you and with every creature that was with you - the birds, the livestock and all the wild animals, all those that came out of the ark with you - every living creature on earth. I establish my covenant with you: Never again will all life be cut off by the waters of a flood; never again will there be a flood to destroy the earth.'" NIV translation

God promised everything on the earth that there would never be a flood like what they had been through.

What a wonderful God! Imagine what it would be like to live through this flood and then to wonder every time it rained whether it was going to happen again. Everything would be living in a constant state of fear and that is not God's desire. 

Genesis 9:12 - 15

"And God said, 'This is the sign of the covenant I am making between you and every living creature with you, a covenant for all generations to come: I have set my rainbow in the clouds, and it will be the sign of the covenant between me and the earth. Whenever I bring clouds over the earth and the rainbow appears in the clouds, I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become a flood to destroy all life.'" NIV translation

God even gave us a sign in the sky to help us to remember His promise. God knew that, over time, the covenant would be forgotten by man and so He puts a rainbow in the sky to remind us of the promise when the rain comes. 

Genesis 9:16 & 17

"'Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth.' So God said to Noah, 'This is the sign of the covenant I have established between me and all life on the earth.'" NIV translation

We cannot understand the mind of God and so we sometimes fear that God will forget things but He does not want us to live in fear of a world wide flood. This promise is repeated again (twice) and the number two is associated with the number of legal witnesses that was required to establish something as truth.