Decide which of the following statements are true, and which are false. Prove the true ones and provide counterexamples for the false ones.
a) If x_n is strictly decreasing and 0 <= x_n < 0.5, then x_n → 0 as n grows.
Answer False: since a function has to be strictly decreasing and bounded below to converge to a given number. The sequence could be decreasing, yet converging to another value.
take x_n = 1/6 + 1/n, the sequence is decreasing yet approaches 1/6, as n grows.
c) If x_n is a strictly increasing sequence and |x_n| < 1 + 1/n for n = 1,2,3,... , then x_n → 1 as n grows.
Can someone verify part a) and Can someone please help me with part c)?