Please write up the following exercises from the text "Introduction to
Analysis, Second Edition" by William Wade, Prentice Hall 1999.
232[3,4,7],
235[1c, 3, 5a].
Here are the equivalent exercises for those using the First Edition:
223[3,4,7],
239[2],
Let x_{k} = ( ln(k+1) - ln k, 2^{-k} ). Using definition 5.9ii,
show that the limit of {x_{k}} exists.
Let X be Euclidean Space.
Show that a sequence in X can have at most one limit.
Show that if {x_{n}} , n=1,2,3,... is a sequence
in X which converges to a and
{x_{nk}}, k=1,2,3,..., is any subsequence
of {x_{n}}, n=1,2,3,...
then x_{nk} converges to a as
k tends to infinity.