How to prove this sum is positive definite

Question

I have a sum that is the norm for $l^2$ (the set of bounded series so $f = (f_n)_{n =1)}^{\infty} | \sum f_i^2 < \infty $). In order to prove that my norm is positive definite I must prove that the following is true:

$$\sum_{i=1}^\infty f_i^2+\frac 1 {8} f_if_{i+1} > 0$$ for $f\neq0$ and the sum being equal to $0$ when $f = 0 $.

I tried making appear $(f_i + f_{i+1})^2$ in order to prove that the sum satisfied the inequality but was unable to

Answer 1

$$\sum_{i=1}^\infty\left(f_i^2+\frac18 f_i f_{i+1}\right)=\frac1{16}f_1^2+\frac78\sum_{i=1}^\infty f_i^2+\frac1{16}\sum_{i=1}^\infty(f_i+f_{i+1})^2.$$