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I'm assuming that your question is about a rocket that starts from rest, so
the question can be restated as "If a rocket starts from rest, can its
speed v ever exceed the speed v_ex of the exhaust relative to the rocket?"
The answer is to this question is "Yes, and by a lot!" (In other words, the
original statement in your message just isn't so.)
The final speed v will exceed the exhaust speed v_ex if the initial
combined mass m_0 of the rocket and the propellant is greater than the
final mass m of the rocket by a factor of at least e = 2.71828 . . ., that
is, if m_0/m > e. As an example, the main engines of the Space Shuttle use
a mixture of liquid hydrogen and liquid oxygen ////
Cheers,
Roger Freedman