​$$f:\N\to\N\ \text{be a bounded function.Which of the}\\\text{following statements is not true?}$$​​

Correct option is C

​$$\text{As } f : \mathbb{N} \to \mathbb{N} \text{ is a bounded function} \\\text{So, } \lim_{n \to \infty} f(n) = K \quad (\text{finite}) \\\text{So, } \lim_{n \to \infty} \big(f(n) + n\big) = K + \infty = \infty \\\liminf_{n \to \infty} \big(f(n) + n\big) = \limsup_{n \to \infty} \big(f(n) + n\big) = \infty \\\text{So, } \liminf_{n \to \infty} \big(f(n) + n\big) \notin \mathbb{N}\\\textbf{For ex:-}\\ Let,\ f(n)=\frac{1}{n}\ \implies\lim_{n \to \infty} f(n) = 0\ But,\\\lim_{n \to \infty}(\frac{1}{n}+n)=\infty \notin\N$$​​