State whether the following statements are True or False. Justify your answers. 
a) The sequence  is convergent in  under the discrete metric on  .
b) A subset in a metric space is compact if it is closed.
c) Continuous image of a path connected space is path connected.
d) The second derivative of a linear map from  to  never vanishes.
e) If  for all  , then .

a) **True.** In the discrete metric, the distance between distinct points is always either 1 or 0. For any point in the sequence, the distance between consecutive terms is always 1, so the sequence converges to each of ________ __________ __________ _____ _________.
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_________ _________ ___ ____ __________ _____ __________ ________ ____.
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_________ __________ _________ _____ __________ ___ ___ ____ _______ ____ ___.
________ ___ _______ _________ ____ _________ _____ ________ ________ __________ ___ _________.
____ _______ _____ ____ ________ _______ ________ ______.
____ _______ ____ ______ _______ ____.
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_______ _________ ____ __________ _______ _____ _________ _____ ______ _______ ___ _____.
_________ __________ ________ _________ ____ ______ __________ _____ _______ ______.
______ ____ _______ ____ _____ _____ __________ ____.
_______ _____ ______ _________ ______.
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