Let $f : [-\pi, \pi] \to \mathbb{R}$ be a continuous $2\pi$-periodic function whose Fourier series is given by$$\frac{a_0}{2}+\sum_{k=1}^{\infty}(a_k \cos kt + b_k \sin kt)$$ Let, for each $n \in \mathbb{N}$, $$f_n(t) = \frac{a_0}{2}+\sum_{k=1}^{n}(a_k \cos kt + b_k \sin kt)$$ and let $f_0$ denote the constant function $\frac{a_0}{2}$. Which of the following statements are true?
(a) $f_n \to f$ uniformly on $[-\pi, \pi]$.
(b) If $\sigma_n = \frac{f_0+f_1+\ldots+f_n}{n+1}$, then $\sigma_n \to f$ uniformly on $[-\pi, \pi]$.
(c) $\int_{-\pi}^{\pi}|f_n(x) - f(x)|^2 dx \to 0$, as $n \to \infty$.
Option a is incorrect. Actual result is that if is $2\pi$-periodic, continuous, and $f'$ is piece-wise continuous, then $f_n$ converges to $f$ uniformly. So to counter this take any $2 \pi$ periodic function whose derivative is not piece-wise continuous. Any counterexample?
How to look for other two?
Your argument for a) is not correct. You are quoting a theorem and using its converse. a) is false and b), c) are true. All three are standard results in the theory of Fourier series and it is not advisable to reproduce the proofs here. A good reference book is Fourier series by Edwards.