Does $\lim_{n\to \infty}\left(\prod_{k=1}^n a_k\right)^{1/n}<1\implies \lim_{n\to \infty}\left(\prod_{k=1}^n\max\{ 1,a_k\}\right)^{1/n}=1 $ I have the positive valued sequence $\{a_n\}$ satisfying the condition $$\lim_{n\to \infty}\left(\prod_{k=1}^n a_k\right)^{1/n}<1$$ Is it true that this implies $$\lim_{n\to \infty}\left(\prod_{k=1}^n\max\{ 1,a_k\}\right)^{1/n}=1 ?$$ Though I am really dubious about this one, I am finding it difficult to prove or disprove this. A possible idea that I have is that I can write $$\max\{1,a_k\}=1+b_k$$ where $b_k\ge 0$ and then possibly find a way to show that $b_k\to 0$ which would serve the purpose, but I am really having problem pursuing this idea. So any help is appreciated. Thanks in advance. 

Following the comments above: faced with infinite products, working with logarithms (and thus the equivalent statements in terms of infinite sums) usually helps.
In this particular case, no matter how you build your intuition (e.g., as above or not), you may end up considering a counterexample such as
$a_{2n} = \frac{n}{3}$, $a_{2n+1}=\frac{2}{n}$.
You do have $$\left(\prod_{k=1}^n a_k\right)^{1/n}\xrightarrow[n\to\infty]{}\frac{2}{3}$$ (e.g., by considering the two subsequences $(\prod_{k=1}^{2n+1} a_k)^{1/{2n+1}}$ and $(\prod_{k=1}^{2n} a_k)^{1/2n}$; but
$$\left(\prod_{k=1}^n \max(1,a_k)\right)^{1/n}\xrightarrow[n\to\infty]{}\infty$$ 
