"By faith Noah, when warned about things not yet seen, in holy fear built an ark to save his family. By his faith he condemned the world and became heir of the righteousness that comes by faith." NIV translation
In the case of Noah, we see that our faith delivers us from the judgment that is coming on this earth. Even though he had never seen rain, he believed God and followed His instructions for preserving his life and the lives of his family. It is the same for us, as Christians, as we believe that God made a way for our salvation and we simply follow His instructions to grab hold of that life.
"By faith Abraham, when called to go to a place he would later receive as his inheritance, obeyed and went, even though he did not know where he was going. By faith he made his home in the promised land like a stranger in a foreign country; he lived in tents, as did Isaac and Jacob, who were heirs with him of the same promise. For he was looking forward to the city with foundations, whose architect and builder is God." NIV translation
The first three examples concerned people before the flood of judgment. Now, we shift to faith in the post-flood era and it begins with Abraham. Through Abraham, we see that true faith leads to obedience as he heard God tell him to go to an unknown place and he did so. There, he lived like a stranger in the land but held onto the promise that God was going to give him the land.
This is much like our walk of faith as Christians where we are called to live as strangers on this earth. We are looking forward to the new heaven and the new earth that is to come just as Abraham was looking forward to ownership of the promised land.
"By faith Abraham, even though he was past age - and Sarah herself was barren - was enabled to become a father because he considered him faithful who had made the promise. And so from this one man, and he as good as dead, came descendants as numerous as the stars in the sky and as countless as the sand on the seashore." NIV translation
The case for faith is based on the character of God. God is faithful meaning He does not go back on promises that He makes. Abraham believed that and was enabled to have children in his old age.
"All these people were still living by faith when they died. They did not receive the things promised; they only saw them and welcomed them from a distance. And they admitted that they were aliens and strangers on earth." NIV translation
This is hard for us to accept today as we are so used to instant gratification. These people waited on the Lord to fulfill His promises in His time. They were able to do this because they looked at things from an eternal perspective instead of our here and now way. The only way that it is possible to have an eternal perspective is through faith in God.