What is the biblical basis for the belief that Satan was the serpent in the Garden of Eden?

Question

This seems to be the generally accepted belief, and it's taught to children in Sunday School in many churches, but is there biblical basis for it? Here's the actual text:

1 Now the serpent was more subtle than any beast of the field which the Lord God had made. And he said unto the woman, Yea, hath God said, Ye shall not eat of every tree of the garden? 2 And the woman said unto the serpent, We may eat of the fruit of the trees of the garden: 3 But of the fruit of the tree which is in the midst of the garden, God hath said, Ye shall not eat of it, neither shall ye touch it, lest ye die. 4 And the serpent said unto the woman, Ye shall not surely die: 5 For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil. 6 And when the woman saw that the tree was good for food, and that it was pleasant to the eyes, and a tree to be desired to make one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat.

—Genesis 3:1-6

What biblical basis is there for this belief? I did see another question addressing Satan being referred to as a serpent a lot in the bible, but then there are verses like John 3:14, where Jesus makes an analogy where he is a serpent:

And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up:

People may say that this is a trivial and unimportant question, but I think it has serious implications on whose fault sin is. If it's Satan, who is still here today, maybe it is as some churches make it out to be, that the only reason anyone sins is because of Satan, and that if he were to be stricken from existence, we would no longer sin. If it wasn't Satan, that leaves open the possibility of us actually voluntarily sinning, back then and today. Thanks in advance :)

(This is the 6,000th question :))

Answer 1

The answer to the question regarding the first book of the Bible actually comes from the last book of the Bible:

And the great dragon was thrown down, the serpent of old who is called the devil and Satan, who deceives the whole world; he was thrown down to the earth, and his angels were thrown down with him. Revelation 12:9 NASB

And he laid hold of the dragon, the serpent of old, who is the devil and Satan, and bound him for a thousand years. Revelation 20:2 NASB

It is understood by many that "the serpent of the old" ties back to Genesis 3, where the serpent deceived the woman, and through her, the entire world.

The Serpent in John 3:14 does not refer back to Genesis 3, but to Numbers 21, which gives the account of the Israelites in the wilderness. God sent a plague of "fiery serpents" to punish the people for their rebellion against Him. The cure for the fashioning of a serpent and raising it up on a pole. Anyone who looked up on the serpent, having been bitten, would be healed. Jesus makes the connection to looking upon Him with faith, as He would be raised up on the Cross. The bronze serpent is a type of Christ and really illustrates the Gospel quite well.

And Moses made a bronze serpent and set it on the standard; and it came about, that if a serpent bit any man, when he looked to the bronze serpent, he lived. Numbers 21:9 NASB