This does not follow. Christ died so that your works may have the fruit of salvation. Without Him, they would be dead, see Romans 3-5. With Him, they brign you to the finish line:
24 Know you not that they that run in the race, all run indeed, but one receiveth the prize? So run that you may obtain. 25 And every one that striveth for the mastery, refraineth himself from all things: and they indeed that they may receive a corruptible crown; but we an incorruptible one. 26 I therefore so run, not as at an uncertainty: I so fight, not as one beating the air: 27 But I chastise my body, and bring it into subjection: lest perhaps, when I have preached to others, I myself should become a castaway.(1 Corinthians 9)
"I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me." -- Galatians 2:20"But by the grace of God I am what I am: and his grace which was bestowed upon me was not in vain; but I laboured more abundantly than they all: yet not I, but the grace of God which was with me." -- 1 Corinthians 15:10
The problem stems from the RC misunderstanding of justification (acquitted once for all the sins of the elect) and sanctification (the ongoing, predestined work of the Holy Spirit to bring the elect to repentance, obedience and faith.)
Note: on the other hand Jesus died on the cross for Something.. WHAT?..
The works we do are evidence that we are saved.
Faith without works is dead (James) - the demons also believe that Jesus Christ is the Son of God - and they tremble.
The bottom line, dear annalex, is that Jesus does not say in the Sermon on the Mount to "do" perfect things or to "say" perfect things - but rather, to "be" perfect.
If we say that we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, he is faithful and just to forgive us [our] sins, and to cleanse us from all unrighteousness. If we say that we have not sinned, we make him a liar, and his word is not in us. - I John 1:5-10 (also John 15)
That, my beloved sisters and brothers, is the Spiritual discernment I have received of John 6 - we are to feed on Him, The Living Word of God. That is the meaning of Lord's command, "take, eat, this is my body" and "Drink ye all of it; For this is my blood of the new testament, which is shed for many for the remission of sins." (Matt 26)