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Given a sequence $\{a_n\}_{n=1}^{\infty}$ and $c>0$ such that $a_{n+1} - a_n>c$ for every $n$, prove $\lim_{n\to\infty} a_n=\infty$.

I proved $a_n$ is monotonic increasing, but I'm having hard time proving it's unbounded.

Any ideas?

3 Answers 3

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From the fact that $a_{n+1} > c+a_n$, you should prove (by induction) that $a_{n+1} > n\cdot c + a_1$.

Once you have that, proving that the sequence is unbounded should be a piece of pie, since for every $M\in\mathbb R$, you can set $n=\frac{M-a_1}{c}$ and get that $a_{n+1} > M.$

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    Yes, I've seen a similar solution, thought there was another way. Could you please explain how did you think of the inequality need to be proven ? – 2017-02-14
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    @Itay4 Well, after you just rearange $a_{n+1} - a_n>c$ into $a_{n+1} > c + a_n$, you can see that this equality actually means that "after each step, you move at least $c$ to the right". That's only one logical leap from thinking OK, so after $2$ steps, I am $2c$ to the right, and after $1000$ steps, I am $1000c$ to the right, so after $n$ steps, I will be $nc$ to the right". – 2017-02-14
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    @Itay4 Then, you just put that into equation form and rigorously prove it. – 2017-02-14
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    Got it, thanks ! – 2017-02-14
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Prove $a_{n+1}>a_1+nc$ using induction or telescoping.

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you can see that

$$ a_n-a_0= \sum_{i=1}^{n } a_i-a_{i-1}> c\sum_{i=1}^{n} 1 =cn $$

ie $$\infty = \lim nc+a_0 < \lim a_n $$ hence $a_n\to \infty$

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    Nice ! I know that weak inequality is preserved in limits, is it the same on strong one? – 2017-02-14
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    What do you mean by weak inequality in this context? Please clarify – 2017-02-16
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    I know that if $a\ \geq b$ then $lima\ \geq lim b$. Is it the same with "$>$" ? – 2017-02-16
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    $a1\implies x< x^2$ but the limits coincide as $x\to 1.$ – 2017-02-17
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    Got it, thanks! – 2017-02-17