Question

Use the definition of limit to prove that the sequence {n −(1/n)}n=1 to infinity is divergent.

Answer 1

we need to show that given M $\gt 0,$ there exists a natural number N(depending on M) such that $(n-\frac{1}{n})n \gt M$ whenever $n \gt N$

now,

$(n-\frac{1}{n})n \gt n-1 \gt M$ whenever $n \gt N = M+1$

therefore, taking N=M,

$(n-\frac{1}{n})n \gt M$ whenever n $\gt N$