In parabolic PDEs, if we apply De Giorgi-Nash-Moser estimate, we usually encounter the following inequality $Q_k \leq \f{2^k}{M}Q_{k-1}^{1+\ep},\qquad \forall\,k\geq 1,$ for some $\ep, M>0$, and we then conclude from this inequality that $Q_k$ is uniformly bounded if $M$ is large enough, which is the goal of this note. **Lemma 1.** Fix $\ep>0$ and let $Q_k$ be a positive sequence such that $Q_k \leq \f{2^k}{M}Q_{k-1}^{1+\ep},\qquad \forall k\geq 1,$ for some $M>0$. Then if $M = Q_0^{\ep/(1+\ep)} 2^{1/\ep},$ then $Q_k \leq \f{M^{1/\ep}}{2^{1/\ep^2}} = \f{Q_0}{ e^{1+\ep} },$ and if $M> Q_0^{\ep/(1+\ep)} 2^{1/\ep}.$ then $Q_k \rightarrow 0,\qquad k\rightarrow +\infty.$ //Proof.// Let $q_k=\ln(Q_k)$ and we have $$(1)\qquad q_k \leq (1+\ep)q_{k-1} + k\ln 2 - \ln M.$$ Iterating $(1)$, we find $$\begin{split} q_k&\leq (1+\ep)q_{k-1} + k\ln 2 - \ln M ~\\ &\leq (1+\ep)\big[ (1+\ep)q_{k-2} + (k-1)\ln 2 - \ln M \big]+ k\ln 2 - \ln M ~\\ &\leq (1+\ep)^2q_{k-2} + (k + (k-1)(1+\ep) )\ln 2 - (1 + (1+\ep))\ln M ~\\ &\leq \cdots ~\\ &\leq (1+\ep)^k q_0 +\big(\sum_{i=0}^k (k-i)(1+\ep)^i \big)\ln 2 - \big(\sum_{i=0}^k (1+\ep)^k\big)\ln M. \end{split}$$ We directly compute $\sum_{i=0}^k (1+\ep)^k = \f{ (1+\ep)^{k+1} - 1}{\ep},\qquad \sum_{i=0}^k (k-i)(1+\ep)^i = \f{ (1+\ep)^{k+1} - 1}{\ep^2}-\f{k+1}{\ep}.$ Hence, we obtain $q_k\leq (1+\ep)^k q_0 + \f{ (1+\ep)^{k+1} - 1}{\ep}\big( \f{\ln 2}{\ep} - \ln M \big),$ which implies that $\begin{split} (2)\qquad Q_k&\leq Q_0^{ (1+\ep)^k } \big(\f{2^{1/\ep}}{M}\big)^{(1+\ep)^k(1+\ep)/\ep} \big(\f{2^{1/\ep}}{M}\big)^{-1/\ep}~\\ &\leq \f{M^{1/\ep}}{2^{1/\ep^2}} \bigg(\f{Q_0 2^{(1+\ep)/\ep^2} }{M^{(1+\ep)/\ep}}\bigg)^{(1+\ep)^k}. \end{split}$ Observing that $Q_0 2^{(1+\ep)/\ep^2} \leq M^{(1+\ep)/\ep} \qquad \Longleftrightarrow\qquad M\geq Q_0^{\ep/(1+\ep)} 2^{1/\ep},$ we conclude from $(2)$ that the claims in **Lemma 1** hold. This ends the proof. $\square$