Bounded operator

I am trying to show that the operator for Hilbert space is bounded. It is given that $\{e_n, n \in N\}$ be an orthonormal basis for Hilbert space .I have started the proof with for $x=\sum_{n\in N}(x|e_n)e_n$. We define $Tx= \sum_{n\in N}(1+\frac{1}{n})(x|e_n)e_n$ , this implies that $\| {Tx}\|^2= \sum_{n\in N}(1+\frac{1}{n})^2|(x|e_n)|^2$. Now I don't know how to proceed to show that this $T$ is bounded.

$\| {Tx}\|^2= \sum_{n\in N}(1+\frac{1}{n})^2|(x|e_n)|^2 \le \sum_{n\in N}4|(x|e_n)|^2=4\sum_{n\in N}|(x|e_n)|^2=4||x||^2$.
• We have $\| \sum_{n = 1}^{\infty} (1 + 1/n) \left< x, e_n \right> e_n \|^2 = \sum_{n = 1}^{\infty} (1 + 1/n)^2 |\left< x , e_n \right> |^2$