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Homework Statement
The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169
Homework Equations
Sup/Inf axioms
Methods of constructing sequences
##ε-N##
##lim(a_n) ≤ sup_n a_n## from question 5 right before it.
I'll split the question into two parts.
The Attempt at a Solution
(a) We must prove every convergent sequence of real numbers is bounded. Suppose ##a_n## is a convergent sequence, that is ##lim(a_n) = L## where ##L \in ℝ##.
So for ## ε = 1, \exists N \space | \space n ≥ N \Rightarrow |a_n - L| < 1##
Thus ##-1 + L < a_n < 1 + L## so that ##a_n## is bounded below by -1 + L and above by 1 + L.
(b) Suppose ##lim(a_n) = L##, we must prove that ##inf_n a_n ≤ L ≤ sup_n a_n##
From question 5 prior, we already know that ##L ≤ sup_n a_n##
So we want to show ##inf_n a_n ≤ L##.
Assume the converse is true, that is suppose that ##inf_n a_n > L##.
Since ##inf_n a_n ≤ sup_n a_n##, we have that ##sup_n a_n ≥ inf_n a_n > L##, but this implies that ##sup_n a_n > L## which is a contradiction.