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A Monotone Sequence Bounded by e

​
n
1
The slope of the black line is less than the slope of the red line, which is less than the slope of the line
y=x
. Then for each
n
we have:
ln1+
1
n
1
n
<
ln1+
1
n+1
1
n+1
<1
, therefore
n
1+
1
n
<
n+1
1+
1
n+1
<e
.
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