Use the definition of limit to prove that the sequence { 1/√n+1 } converges to 0
The sequence (1n+1)n\left(\frac{1}{\sqrt{n}+1}\right)_n(n+11)n converges to 0 if
Let ε>0\varepsilon>0ε>0. If we take N=⌈1ε⌉2N=\left\lceil\dfrac{1}{\varepsilon}\right\rceil^2N=⌈ε1⌉2 we obtain N=⌈1ε⌉≥1ε\sqrt{N}=\left\lceil\dfrac{1}{\varepsilon}\right\rceil\geq\dfrac{1}{\varepsilon}N=⌈ε1⌉≥ε1. Then 1N≤ε\dfrac{1}{\sqrt{N}}\leq\varepsilonN1≤ε. Therefore
So, for all n≥N,n\geq N,n≥N,
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