It is known that for integers $n \geq 1$,
$\lim_{ n \to \infty} (1 + n)^{1/n} = e = 2.718\dots$
For integer $N \ge n$, is it true that:
$\lim_{ n, N \to \infty} (1 + n + N)^{1/n} > e\ \ ?$
Suppose the sequence is monotone either increasing or decreasing and also that infinitely many terms of the limit
$\lim_{ n, N \to \infty} (1 + n + N)^{1/n}$
are bounded within some compact interval $[a, b]$ on the real line. Is this limit
$\lim_{ n, N \to \infty }(1 + n + N)^{1/n}$
finite on $[a, b]$? Does it converge to some finite value?
If anyone can help to solve this question then I thank you in advance.