\( \Large a + \frac{1}{b} = 1 \)

=> \( \Large a = \left(1 - \frac{1}{b}\right) = \frac{b - 1}{b} \)

=> \( \Large \frac{1}{a} = \frac{b}{ \left(b - 1\right) } \) [reciprocal]

and \( \Large b + \frac{1}{c} = 1 \)

=> \( \Large \frac{1}{c} = \left(1 - b\right) \)

=> \( \Large c = \frac{1}{ \left(1 - b\right) } \)

Therefore, \( \Large c + \frac{1}{a} = \frac{1}{ \left(1 - b\right)} + \frac{b}{ \left(b - 1\right)  } \)

= \( \Large \frac{1}{ \left(1 - b\right)} - \frac{b}{ \left(1 - b\right)  } \)

= \( \Large \frac{ \left(1 - b\right) }{ \left(1 - b\right) } = 1 \)