What does the Nurse do to betray Juliet in Romeo and Juliet?

In Romeo and Juliet, by William
Shakespeare, the Nurse, who has been so dedicated to Juliet and so supportive of Romeo
in the past, urges Juliet to leave Romeo, and in doing so, loses Juliet's
trust.

One of the reasons that the Nurse may turn her back
on Romeo, and therefore Juliet, is that Romeo killed Tybalt who was a favorite of the
Nurse. It does not matter that Tybalt went looking for a fight, and not only killed
Romeo's dear friend Mercutio, but did so dishonorably. The Nurse blames Romeo for
Tybalt's death.

However, perhaps more than this, the Nurse
is also trying to be realistic.  Juliet's parents have threatened to throw Juliet into
the street if she does not marry Paris. (On a personal level, it is possible that the
Nurse would fear for her own livelihood, as she has been Juliet's nurse her entire life,
and would go with her should she marry, but have no income—with her husband dead—should
Juliet be exiled from her home.) And although the Nurse is generally a foolish kind of
woman, she is fiercely loyal to Juliet. Her reasoning here is solid (especially for
women of that era).

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Romeo is banished, and all the world to nothing
/ That he dares ne'er come back to challenge you. / Or if he do, it needs must be by
stealth. / Then since the case so stands as now it doth, / I think it best you married
with the County.   (III, v,
214-218)

The Nurse then tries
to convince Juliet by listing the positive things about Paris that would make him a good
husband.

Juliet is young and desperately in love. She can
see nothing but her love for Romeo. If she were able to be more objective, she might
have seen that the Nurse was only looking out for Juliet's best interests. However, had
this happened, from the standpoint of the plot, the Nurse would not have been shut away
from their secret plans, and the tragedy of the lovers' last days would not have played
out as Shakespeare intended.