I'm writing a school paper about convergence. I am a bit stuck on proving pointwise convergence by definition for a particular function sequence related to the delta function:
Let $T_n(x)$ be a function sequence defined on a closed interval $[0,1]$ with following properties: $$\text{1.} \ T_n(x) \geq 0, \ T_n \ \text{is continuous.}$$ $$\text{2.} \ \lim_{n\to\infty} \operatorname{supp} (T_n) = \{0\}.$$ $$ \text{3.} \ \int_{-\infty}^{\infty} T_n = 1. $$
It is fairly obvious (I think) that this sequence does converge pointwise to $0$, and I have actually proven it by definition with my teacher a week ago and understood the proof well, but now I'm hardstuck on recreating it for some reason.
To be precise, I need the proof by this kind of definition:
Let ${f_n}$ be a sequence of functions defined on a metric space $X$. Then ${f_n}$ converges pointwise to a function limit $f$ if and only if $$ \forall \ x \in X, \ \forall \ \epsilon > 0, \ \exists \ M > 0 $$ $$ \text{such that} \ |f_n(x) - f(x)| < \epsilon \ \text{when} \ n > M.$$
I would appreciate a hint, a starting point (mostly can't determine which property of the function sequence I've defined can be translated into the $\epsilon$, $M$, $x$, $n$ world) or a complete proof. I already have a decent grasp of proving pointwise convergence for function sequences that are defined explicitly (by a formula).
Thanks in advance for your help.