Let $ {a_k}$ be an unbounded,strictly increasing  sequence of  positive real numbers and $x_k=\dfrac{a_{k+1}-a_k}{a_{k+1}}$. Prove that $\sum_{k=m}^n x_k&gt;1-\dfrac{a_m}{a_n}\forall n\geq m $ and $\sum_{k=1}^\infty x_k$ diverges to $+\infty.$  What I thought: $\sum_{k=m}^n x_k=\sum_{k=m}^n(1-\dfrac{a_k}{a_{k+1}})$  As $0&lt;\dfrac{a_k}{a_{k+1}}&lt;1$,taking $\dfrac{a_k}{a_{k+1}}=1-\dfrac{1}{k}$ for each $m\leq k \leq n$.  But i cant carry it more .Please help me.
Let $\{a_n\}$ be an unbounded, strictly increasing sequence of positive real numbers and let $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$. Which of the following statements is/are correct? (CSIR NET December 2014)         For all $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;1-\bf{\frac{a_m}{a_n}}$   There exist $n\geq m, \sum\limits^{n}_{k=m}x_k&gt;\frac{1}{2}$      My attempt:  Analysing question in light of $a_k=k\implies x_k=\frac{k+1-k}{k+1}=\frac1{k+1}$, I feel both are correct, I doubt if  I can generalise the result in light of this one example.   Any way to prove that this true for general case?    Note: Please do not confuse  with my question. The first two options are different, I've marked the difference in bold.
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